Right now we’ll use this problem to obtain another characterization of right triangles, in addition to the ones here.

## Stretched image

To start with, here’s an enlarged version of the above image:

In any triangle, the ratio of the length of an internal altitude to the “length” of a right bisector is always sandwiched between and .

## Strict inequalities

In general, the ratio under consideration lies strictly between and .

Details here.

Details here.

Details here.

Together we have: .

## Special instance

When the ratio under consideration equals .

Use similar triangles. Or else use the fact that: from example 1. Take so that . Simplify to get .

Details here.

## Takeaway

Let be a triangle in which is the foot of the internal altitude from vertex , is the midpoint of side , and the right bisector of intersects (or , depending on which is greater) at . Then the following statements are *equivalent*:

- is a right triangle in which
- .

## Tasks

- (Silver ratio) Suppose that is a triangle containing an obtuse angle . Let be the foot of the altitude from and let the right bisector of intersect at .
- PROVE that , where is the midpoint of .
- Find a triangle for which , the silver ratio.