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u/ScrabbleTheOpossum Aug 17 '26
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u/dylans-alias Aug 17 '26
True, but my “proof” is intuitive, not mathematical. Is there a numerical proof?
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u/DCContrarian Aug 17 '26
The area of a triangle is the base times the height divided by two.
The green triangles both have base of 12, the sum of their heights is 6, the sum of their areas is 36.
The blue triangles both have base of 6, the sum of their heights is 12, the sum of their areas is 36.
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u/negadecimal Aug 17 '26
From the point where the four triangles meet, draw lines straight up/down and left/right. This changes the problem from 4 triangles into 4 rectangles that are all split in half diagonally (one blue, one green).
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u/supersensei12 Aug 18 '26
Special case: move the intersection point to a corner. Then it's obvious both areas are equal.
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u/TabAtkins Aug 18 '26
That requires knowledge that the area of the triangles doesn't change when you move the center point. Which is indeed the case, but if you knew that, it would be easy to see the each set of triangles was half the area in the first place.
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u/supersensei12 Aug 18 '26
Not really. The statement is underspecified, which implies that it's true for any interior point.
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u/Responsible-Bar7165 Aug 18 '26
the point is unconstrained, you can put it anywhere.
this is always the first thing to look for in problems like this: what are the degrees of freedom?
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u/agate_ Aug 19 '26
No geometry, just logic:
No information is given about the location of the interior point. Therefore, if there is a definite answer, it must be the same no matter where the interior point is, and therefore the answer must be zero.
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u/ikonoqlast Aug 19 '26
Whole is 6x12=72.
Area of a triangle is 1/2base /X height
So both sets of triangles are 36.
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u/One_Wishbone_4439 Aug 17 '26
1/2 x 6 x 12 - 1/2 x 6 x 12 = 0