Lesson Angle bisector drawn to the hypotenuse of a right triangle

Angle bisector drawn to the hypotenuse of a right triangle

This lesson focuses on angles in a right triangle.

Problem
In a right triangle, the angle bisector, the altitude and the median are drawn from the right angle vertex to the hypotenuse.
Prove that the angle bisector of the right angle bisects the angle between the altitude and the median too.

Solution Figure 1 shows the right triangle ABC. The angle ACB is the right              angle. The altitude AD (red line), the angle bisector AE (blue line) and the median AF (green line) are drawn from the right angle vertex (point C) to the hypotenuse AB. We need to prove that the angle bisector CE of the right angle ACB bisects the angle DCF too. The proof is very simple (Figure 2). Let be the angle ABC and be the angle BAC. In the right triangle ABC the angle BAC is complement to the angle ABC, therefore = 90° - (see the lesson Angles basics under the topic Angles, complementary, supplementary angles of the section Geometry in this site).

Figure 1. The angle bisector, the altitude                 and the median of a right triangle

Figure 2. To the solution of the Problem

In the right triangle ADC the angle ACD is complement to the angle CAD. Therefore, the angle measure of the angle ACD is equal to 90° - = 90° - (90° - ) = .
So, the angle ACD is congruent to the angle ABC. This is why in Figure 2 we denoted the angle ACD by the same symbol as the angle ABC.

From the other side, the median in a right triangle divides it in two isosceles triangles. This fact was proved in the lesson Median in a right triangle drawn to its hypotenuse
under the current topic Geometry of the section Word problems in this site.
In our case, this means that the triangle BFC is the isosceles triangle. Hence, the angles FBC and FCB are congruent and both have the same angle measure .

The rest of the proof is straightforward.
Since the segment CE is the angle bisector of the right angle ACB, the angles ACE and BCE have the angle measure 45° each.
The angle DCE has the angle measure 45° - . The angle ECF has the angle measure 45° - too.
Since the angles DCE and ECF have the same angle measures, they are congruent. Hence, the segment CE bisects the angle DCF. The proof is completed.