Feyman's 1/7 Area Triangle: A Beautiful New Proof By Dissection
Monday 30 March 2026 at 19:49
Visible to anyone in the world
Edited by Richard Walker, Monday 30 March 2026 at 20:37
Claim: In any triangle ABC, if we draw a line from each vertex to the point along on the side opposite, the inner triangle so formed has area equal to of the area of the orginal triangle.
I posted a proof of this on 28 March but since then I've thought of a simpler and far nicer proof.
Figure 1
In Figure 1 triangle ABC, R, S and T are trisection points of the sides they lie on; and K, L and M the midpoints of the sides. XYZ is the Feynman "one-seventh" triangle formed by lines AT, BS and CR.
In Figure 2 we have rotated triangles XBL, YCM and ZAK through about the midpoints L, M and N respectively.
Figure 2
Figure 2 shows triangle ABC can be dissected into a figure consisting of seven congruent triangles, all of equal area, and that the Feynman triangle XYZ, numbered 7 in the diagram, is one of them. It follows immediately that its area is the area of triangle ABC.
Of course, like the proof I gave previously, this must have already been discovered by many before me, but I worked it out for myself and the 'εὕρηκα' moment was very pleasing.
Feyman's 1/7 Area Triangle: A Beautiful New Proof By Dissection
I posted a proof of this on 28 March but since then I've thought of a simpler and far nicer proof.
In Figure 1 triangle ABC, R, S and T are trisection points of the sides they lie on; and K, L and M the midpoints of the sides. XYZ is the Feynman "one-seventh" triangle formed by lines AT, BS and CR.
In Figure 2 we have rotated triangles XBL, YCM and ZAK through about the midpoints L, M and N respectively.
Figure 2 shows triangle ABC can be dissected into a figure consisting of seven congruent triangles, all of equal area, and that the Feynman triangle XYZ, numbered 7 in the diagram, is one of them. It follows immediately that its area is the area of triangle ABC.
Of course, like the proof I gave previously, this must have already been discovered by many before me, but I worked it out for myself and the 'εὕρηκα' moment was very pleasing.