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Here is a good diagram that pictorially describes what I have been talking about with the "angle between the sun and the moon at the phase of quarter moon."


This is how it looks during the First Quarter moon, which will be coming up in about 2 weeks.
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We just had the Last Quarter moon last week, on Thursday. I measured the angle between the sun and the moon and found it to be greater than 89 degrees and less than 90 degrees.
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There are consequences to these measurements, since as you can see in the diagram, the three corners of the geometrical figure described by the earth, moon and sun is a right triangle.
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Right triangles are very easy to use in determining proportions between the lengths of the sides when one of the not-square interior angles is known. We already know that the angle at the moon is 90 degrees because it is a quarter moon, which see. 
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We have measured one of the other two angles, the angle from earth between the sun and moon, and have found it to be consistently close to 90 degrees at the quarter moon phase.
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Therefore, without having to travel to the moon or the sun to check the angles of the other two bodies, we can surmise what the angle between the earth and moon as viewed from the sun would be.
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Since any triangle has 3 interior angles and their sum is always 180 degrees, we subtract the 90 degrees at the moon and the 89-plus degrees at the earth and come up with an angle of less than one degree at the sun.
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ETC............

Great Circle Sydney to Santiago on "Google Earth"
   about 11,400 km

Shortest Sydney to Santiago on "Gleason Map"
   about 25,400 km

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There have been commercial flights between Sydney and Santiago -- they've had a duration and mileage comparable with the former (11,400 km) but not with the latter (25,400 km).
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Again, no flat-earthers to reply:  they'd have to say, NO THATS(sic)NOT TRUE, or something like that.
(Leaving out the apostrophe, of course.)
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Great Circle route flights between Perth, Australia and Buenos Aires, Argentina would take the plane almost directly over the South Pole:
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The alternative route on the bottom appears obviously curved here - between Buenos Aires and Sydney. That's a Great Circle route too, but our angle of view is not right above its coordinates. This view is directly above the South Pole, so the top route between Perth and Buenos Aires appears only very slightly curved.
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If you understand what Great Circle route is, you'll understand the difference and reasons. If you don't, you won't. And if you don't want to, you NEVER WILL.
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Simple.
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Scripture is clear for those who read it, earth is flat.  The Church teaches it. Science confirms it. Only the ill-informed, blind or confirmed globalist/pagan denies it.  There is not one single teaching that the earth is a moving ball.  Nor that it is a hanging ball in mid-air. Scripture says the earth has a foundation.  A firm foundation.  Pillars.  Bounds. A firmament that divides the corporal on earth from the spiritual in heaven above, and also within that hardened dome, the sun, moon and stars move.  Scripture teaches that stars have fallen to earth and that stars will fall to earth, an absurd notion to those who believe stars are hundreds of times larger than earth and made of earthly elements like rock. The saints teach there is a center on earth, which is Jerusalem. They teach there are no people walking upside down relative to the rest.  These are Catholic teachings, and cannot be denied without demonstrating a loss of Faith in truth.  Those who persist in the fallacy of a ball earth are deniers of God and His Word.

  

Well said. Unfortunately, there are those Catholics who seem to believe that the Old Testament is not accurate, and that science has replaced and corrected the OT in regards to the form of the earth.

Yes, it is unfortunate that Catholics cannot read scripture and know that what it says is gospel. They play the Protestant game of not understanding the words because they are in love with their own notions.  The same Catholics will get into wars over the founding of the papacy passages or Jesus on the Blessed Sacrament because scripture says what it means, then say that scripture isn't saying what it means when is speaks of the four corners of the earth, or the firmament, or the bounds on the waters, the pillars, the foundation of earth, or the celestial realm under the dome.  For this ridiculous kind of equivocation there is little excuse. They are the type of Catholics who refuse to learn unless the Pope converts and makes a papal pronouncement on it. Of course, not even that would work for the naysayers as scripture testifies so clearly: Luke 16:31   And he said unto him, If they hear not Moses and the prophets, neither will they be persuaded, though one rise from the dead. 

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See how flat-earthers run away scared whenever you post a little geometry?
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Flat-earthism has nothing to do with observed facts and verifiable reality.
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It's all subjectivism, rumor and innuendo of the willfully ignorant.
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Sorry, looks like flat-earthers flee to their hackneyed misinterpretation of Scripture.
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What does Scripture have to say about the solubility of amino acids?
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If you are a doctor treating a patient for illness do you go to the Bible to find out about God's revelation concerning the solubility of amino acids?
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I didn't post this week about the first quarter moon, but it happened anyway. Imagine that!
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The moon doesn't wait for Internet posts before it changes phases.
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The moon had its first quarter again, at 7:48 pm Tuesday June 2nd, as scheduled on the Old Farmer's Almanac website -- a place where the words "flat earth" have never appeared because it's not a ridiculous publication.
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Therefore, farmers all over America have been purchasing and using a publication for well over 200 years (since 1792) that has never mentioned a "flat earth." But what do farmers know, anyway? 
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But I digress. The moon rose, as predicted, and I measured the angle between it and the sun just before the quarter moon phase and just after.  It started at less than 88 degrees (waxing crescent moon), became Quarter Moon at less than 90 degrees and greater than 89 degrees, and finished at greater than 90 degrees (waxing gibbous moon).
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This is the same results I got last month. Surprise, surprise. I expect it's going to be the same every time I check it, until the cows come home (or the sheep and goats if you're lactose intolerant).
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This figure of 89-plus degrees is very interesting. It is always less than 90 degrees, which is comforting, because it actually makes sense this way. 
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Triangles have the consistent property of having three interior angles that add up to 180 degrees. I know, I know, that's witchcraft HAHAHAHAHA.  NOT.
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When the moon is at just one quarter (a state that lasts for a minute or two but who's counting that closely?), the angle as observed from the moon between the earth and sun is just 90 degrees, which is why we can see the moon appearing as a Quarter Moon. That means the earth, moon and sun (for a minute or two anyway) are located just at the points of a right triangle. 
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Right triangles are great because trig functions are very easy to use with them.
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Therefore, we have one known angle, 90 degrees between earth and sun as viewed from the moon, and we didn't have to travel to the moon to know this.
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And we have a second known angle, the angle between the moon and sun as viewed from earth, which is where we are so we didn't have to travel anywhere.
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Consequently we can calculate the third angle, namely, the angle between the earth and moon as viewed from the sun, and we didn't have to travel to the sun (fortunately).
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180 - 90 - 89.5 = 0.5 degrees.  But we do not have an accurate number, only an approximation, since our instruments are crude. We know the angle between sun and moon is greater than 89 and less than 90 so I have approximated it at 89.5 here. If that is a good number, then the angle between earth and moon as viewed from the sun would be one half of one degree.
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More witchcraft!! 
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Burn him at the stake!! 
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Apostate!! Heretic!! Calumniator!! Ingrate!! Malefactor!!
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Curiously, all the accusations against me for being "uncharitable" toward flat-earthers, the flat-earthers practice toward me without restraint. Interesting. That could be called duplicity and hypocrisy, but we can't say that because it's "uncharitable." Correct?
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Every day, the moon proclaims in a loud voice (for those with ears to hear and eyes to see) that the earth is a globe.
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The phase of the moon observed in all parts of the world are exactly the same at any time.
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Whether you are in Nebraska or Argentina, Hawaii or South Africa, Antarctica or Australia, China or France, Greenland or Canada, the phase of the moon at any moment is the same from all over the earth.
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If the earth were "flat" that would not be the case, and depending on where you are on the earth you would see various phases of the moon because your angle of view and the sun's angle to the moon would vary depending on your position. But that's not what happens. Never has, and it never will.
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In fact, when the moon reaches its first quarter (as it did just last week), the time when the quarter moon occurs is the same all over the earth. The moon reaches its quarter phase at the same moment regardless of where the observer is on earth, but that would not be the case if the earth were "flat" with the sun and moon twirling around in inexplicable silly spirals above it like flies on (redacted) on a dinner plate.  
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5:43 am in California (UTC-7) is 8:43 am in New York and it's 12:43 pm in London, England. Therefore, the First Quarter moon occurred just (43 minutes) after noon in London, at the same moment when it occurred in California. If the earth were "flat," that would not be the case as the moon's phases would occur at different times in different parts of the world. But they don't, so the earth is not "flat."
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The earth is not "flat" and the earth is at great distance from the moon which is why the moon appears with the same phase all over the earth every day, 24 hours a day, 7 days a week, 12 months a year, even on leap years.
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The moon appears as it does for all to see (those with eyes to see, that is), exactly the same at all times, because the moon is very far away, where it has the same appearance from great distance. If the moon were only a few thousand miles from earth (as the flat-earthers falsely claim) it would not appear the same to people all over the earth at all times and in all ages.
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The evidence is everywhere you look, if you bother to  open your eyes and take a look.
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This is not news. It's been going on for centuries.


http://www.public.asu.edu/~atpcs/atpcs/Univ10e/chapter03-06.html

Chapter 3-6: Ancient astronomers measured the size of the Earth and attempted to determine distances to the Sun and Moon

Ancient (Greek) astronomers had a good knowledge of the shapes, sizes and distances of the Earth, Moon, and Sun
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They knew the Earth is spherical because

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The Earth's shadow (apparent during lunar eclipses) is always circular

On the open sea, only the tops of distant ships are visible - the lower portions are below the horizon (i.e., the ships don't just appear smaller)

Different parts of the celestial sphere are visible if one travels north or south

They knew the Moon is spherical because
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This is the easiest way to explain the sequence of lunar phases

Eratosthenes determined the size of the Earth by measuring the angle of the noontime Sun from different locations on Earth
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At a well in Syene, the Sun is at the zenith at noon on the beginning of summer
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But an obelisk at Alexandria, the Sun casts a short shadow - indicating the Sun is about 7.2 degrees away from the zenith
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The difference in angle (zero versus 7.2 degrees) is a measure of the difference in latitude (using today's terminology)
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Therefore, the linear distance between Syene and Alexandria - about 5000 stades - is 7.2/360 of the Earth's circuмference
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Using our best guess of the length of a stade, Erathosthenes' value for the size (diameter) of the Earth is accurate to a few percent!

From this, one can estimate the size of the Moon
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Observations of lunar eclipses show that the diameter Moon is about 3/8 that of Earth
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Hence, the ancient astronomers could calculate a reasonable estimate of the Moon's size

From this, one can estimate the distance to the Moon
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The Moon has an angular size of 1/2 degree
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Hence, knowing the angular size and the true (physical) size, the ancient astronomers could calculate a reasonable estimate of the distance to the Moon by applying the small-angle formula

Aristarchus determined the distance to the Sun by trying to measure the angle between the Sun  and Moon (as seen by an observer on Earth) at the time of the first quarter (or last quarter) moon
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This angle is close to - but slightly less than - 90 degrees and is very difficult to measure
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A large value means the Sun is far away, while a small one means the Sun is close
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Aristarchus got an angle that is too small and estimated the Sun is only about 20 times further than the Moon
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This also yields an estimated size of the Sun that is 20 times larger than the Moon
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The correct values are 400 not 20, but the technique showed that the Sun is the largest of these objects!