Are All Triangles Isosceles?

Strategies and Hints

  1. Students can prove the theorem false by drawing one triangle that is not isosceles.

  2. AP does not bisect angle BAC. As a result, segments and angles that depend on AP are also "off."

  3. Point P is not in the interior of the scalene triangle. If the triangle is isosceles with AB = AC, the bisector of BAC and the perpendicular bisector of do not intersect in a single point P. Instead they coincide.

  4. Step 5. It is not true that AR + RB = AS + SC.

Solution

The fallacy in the proof lies in the assumption that P lies in the interior of the triangle. If the triangle is not isosceles, P will be outside the triangle as shown in the drawing above. Thus, it will not be true that AR + RB = AS + SC . In the figure above,
AR - RB = AS + SC.