## Teaser 2922: Inscribed

**From The Sunday Times, 23rd September 2018** [link]

The Republic of Mathematica has an unusual rectangular flag. It measures 120 cm by 60 cm and has a red background. It features a green triangle and a white circle. All the lengths of the sides of the triangle and the radius of the circle (which touches all three sides of the triangle) are whole numbers of cm. Also, the distances from the vertices of the triangle to the centre of the circle are whole numbers of cm. The flag has a line of symmetry.

What is the length of the shortest side of the triangle?

[teaser2922]

## Jim Randell 8:29 pm

on19 April 2019 Permalink |The flag has a line of symmetry, so I supposed the triangle was isosceles, with sides

a, a, b.If we split it in half down the middle we get a right-angled triangle, with base

q = b/2, heighthand hypotenusea.The height

hcan be split intorthe radius of the incircle (an integer) plusxthe distance from the incentre to a vertex of the triangle (also an integer), so is itself an integer.The right hand side is the difference of two integers, so is an integer, and hence

b/2is an integer.So

(q, h, a)is a Pythagorean triple.The inradius

rcan be calculated as the area of the isosceles triangle, divided by its semiperimeter.And this must be an integer.

We can the calculate the distance

yfrom the incentre to the lower vertices:And this also must be an integer, so

(r, q, y)is also a Pythagorean triple.This Python program runs in 69ms.

Run:[ @repl.it ]Solution:The shortest side of the triangle is 56 cm.The isosceles triangle has sides measuring 100 cm, 100 cm, 56 cm.

The radius of the incircle is 21 cm. The upper vertex of the triangle is 75 cm from the incentre, the lower vertices are 35 cm from the incentre.

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