PMATH 320: Euclidean Geometry Assignment 2

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PMATH 320: Euclidean Geometry Assignment 2

Due: Thursday May 23 2024 at 11:59pm on Crowdmark

Question 1. Let △ABC be a triangle of perimeter p. Show that the sum of the lengths of the medians of the triangle lies in the interval [4/3 p, p].

Hint: Use the triangle inequality and the fact that the medians all intersect at a points G.

Question 2. Use Heron’s formula to

a) compute the area of a triangle with side lengths 12,13 and 23.

b) Find a symmetric polynomial f(a, b, c) such that for a triangle with side lengths a, b, c and area  one has

Question 3. Let △ABC and △A′B′C ′ be similar triangles.

a) Show that

b) Use part a) to prove the Pythagorean Theorem.

Hint: Make a line from the vertex with the right angle perpendicular to the opposite side and notice some similar triangles.

Question 4. Let △ABS be a triangle with side lengths a, b, c. Let ra, rb, rc be the radii of the excircles and let S be the semiperimeter.

a) Show that

Area(△ABC) = (S − a) · ra = (S − b) · rb = (S − c) · rc

b) Let r be the radius of the incircle of △ABC. Show that

and