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1. Prove that if I and G are the same points, then ABC is an equilateral triangle. 2. The Fermat point: Construct equilateral triangles on each side of a triangle AABC. Prove that the circumcircles of these three equilateral triangles intersect in a common point. Let us call this point F. Prove that if F is inside the triangle, then B A IL LJ ZAFB = ZBFC = ZCFA = 2π/3. When F is inside the triangle, it is called the Fermat point. When F is outside the triangle, the Fermat point is the vertex with the obtuse angle. The Fermat point is not a triangle center, though F is. 3. For the equilateral triangles in Problem 2, let A' be the third vertex of the equilateral triangle with side BC. Prove that AA' passes through F.

1. Prove that if I and G are the same points, then ABC is an equilateral triangle. 2. The Fermat point: Construct equilateral triangles on each side of a triangle AABC. Prove that the circumcircles of these three equilateral triangles intersect in a common point. Let us call this point F. Prove that if F is inside the triangle, then B A IL LJ ZAFB = ZBFC = ZCFA = 2π/3. When F is inside the triangle, it is called the Fermat point. When F is outside the triangle, the Fermat point is the vertex with the obtuse angle. The Fermat point is not a triangle center, though F is. 3. For the equilateral triangles in Problem 2, let A' be the third vertex of the equilateral triangle with side BC. Prove that AA' passes through F.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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1. Prove that if I and G are the same points, then ABC is an equilateral triangle. 2. The Fermat point: Construct equilateral triangles on each side of a triangle AABC. Prove that the circumcircles of these three equilateral triangles intersect in a common point. Let us call this point F. Prove that if F is inside the triangle, then B A IL LJ ZAFB = ZBFC = ZCFA = 2π/3. When F is inside the triangle, it is called the Fermat point. When F is outside the triangle, the Fermat point is the vertex with the obtuse angle. The Fermat point is not a triangle center, though F is. 3. For the equilateral triangles in Problem 2, let A' be the third vertex of the equilateral triangle with side BC. Prove that AA' passes through F.
Transcribed Image Text:1. Prove that if I and G are the same points, then ABC is an equilateral triangle. 2. The Fermat point: Construct equilateral triangles on each side of a triangle AABC. Prove that the circumcircles of these three equilateral triangles intersect in a common point. Let us call this point F. Prove that if F is inside the triangle, then B A IL LJ ZAFB = ZBFC = ZCFA = 2π/3. When F is inside the triangle, it is called the Fermat point. When F is outside the triangle, the Fermat point is the vertex with the obtuse angle. The Fermat point is not a triangle center, though F is. 3. For the equilateral triangles in Problem 2, let A' be the third vertex of the equilateral triangle with side BC. Prove that AA' passes through F.
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