Question

Javier wants to inscribe a circle inside of the triangle ABC shown in the following figure. He begins by constructing the angle bisectors of angles A and B and finding their intersection at point D.
If Javier is trying to inscribe a circle inside of triangle ABC in as few steps as possible, which of the following steps would be the best step for him to take next?

1) Construct the perpendicular line from point D to the line BC.

2) Find the altitude from point C.

3) Use the distance from D to any of the triangle's vertices to set the width of the compass.

4) Construct the angle bisector of angle C.


Thank you :)

Answer 1

Point D, as shown i the figure, is the intersection of the angle bisectors. This point is the Incircle, or the center of the inscribed circle. 

All 3 angle bisectors meet at D, so drawing the angle bisector of C is useless. (Thus step 4 is not the one).

Since we have the center of the inscribed circle, we want to open the compass so that it touches all 3 sides at one point only, that is, we want the 3 sides to be tangent to this circle.

The segments joining the tangency points and D are 3 radii of the circle. We know that a radius is perpendicular to the tangent it touches. 


Thus, we need to draw an altitude from D to any of the sides. 

Answer: 1