A given convex pentagon $ ABCDE$ has the property that the area of each of five triangles $ ABC, BCD, CDE, DEA$, and $ EAB$ is unity (equal to 1). Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.
Problem
Source: 1972 USAMO Problem 5
Tags: geometry, parallelogram
06.03.2010 06:13
29.06.2011 00:32
Interesting! The Bangladeshians (people from Bangladesh) used similar problem for their Mathematical Olympiad 2012 competition: "In a given pentagon ABCDE, triangles ABC, BCD, CDE, DEA and EAB all have the same area. The lines AC and AD intersect BE at points M and N. Prove that BM = EN." Let me suggest these similar problems: 1. Given a convex pentagon ABCDE with M, N, R, S and T be the intersections of AC and BE, AD and BE, AD and CE, BD and CE, BD and AC, respectively such that the areas of triangles ABM, BCT, CDS, DER and AEN are also the same. Prove that ABCDE is a regular pentagon. 2. Given a convex pentagon ABCDE, the areas of triangles ABC, BCD, CDE, DEA and ABE are all the same. Let M, N, R, S and T be the intersections of AC and BE, AD and BE, AD and CE, BD and CE, BD and AC, respectively. Prove that the areas of triangles ABM, BCT, CDS, DER and AEN are also the same.
03.12.2022 09:16
redacted
03.12.2022 10:20
wxl18 wrote:
I think you may have posted on the wrong topic.
13.04.2023 05:18
Vo Duc Dien wrote: Interesting! The Bangladeshians (people from Bangladesh) used similar problem for their Mathematical Olympiad 2012 competition: "In a given pentagon ABCDE, triangles ABC, BCD, CDE, DEA and EAB all have the same area. The lines AC and AD intersect BE at points M and N. Prove that BM = EN." Let me suggest these similar problems: 1. Given a convex pentagon ABCDE with M, N, R, S and T be the intersections of AC and BE, AD and BE, AD and CE, BD and CE, BD and AC, respectively such that the areas of triangles ABM, BCT, CDS, DER and AEN are also the same. Prove that ABCDE is a regular pentagon. 2. Given a convex pentagon ABCDE, the areas of triangles ABC, BCD, CDE, DEA and ABE are all the same. Let M, N, R, S and T be the intersections of AC and BE, AD and BE, AD and CE, BD and CE, BD and AC, respectively. Prove that the areas of triangles ABM, BCT, CDS, DER and AEN are also the same. Cool, are you the creator of these problems? How did you post used a similar problem for 2012 Olympiad when your post was in 2011!? Anyways, back to the problem. futuremospper wrote:
sorry, what was point P?