Baltic Way 1992 · Problem 20

Geometry

Let $a \leq b \leq c$ be the sides of a right triangle, and let $2 p$ be its perimeter. Show that

p(p-c)=(p-a)(p-b)=S

where $S$ is the area of the triangle.

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Review

Angles and distances

Solution

Solution:

By straightforward computation, we find:

\begin{aligned} & p(p-c) = \frac{1}{4}\left((a+b)^2 - c^2\right) = \frac{ab}{2} = S, \\ & (p-a)(p-b) = \frac{1}{4}\left(c^2 - (a-b)^2\right) = \frac{ab}{2} = S. \end{aligned}

Results from Baltic Way 1992

8 teams

Score distribution

All team scores