Question

The angle bisector of angle BAC is a ray AD such that AD is between AB and AC and the measure of angle BAD equal the measure of angle DAC. The distance from a point to a line is measured along a perpendicular. Prove the following theorem. Pointwise characterization of angle bisector. A point P lies on the bisector of angle BAC if and only if P is in the interior of angle BAC and the distance from P to AB equals the distance from P to AC.

Answer 1

The Pointwise Characterization of Angle Bisector theorem is proven by showing that if a point lies on the bisector, the distances from the point to the sides of the angle are equal, and conversely, if the distances are equal and the point is in the interior, then it lies on the bisector.

Step-by-step explanation:

The Pointwise Characterization of Angle Bisector theorem states: A point P lies on the bisector of angle BAC if and only if P is in the interior of angle BAC and the distance from P to AB equals the distance from P to AC. To prove this, first assume that P lies on the bisector of angle BAC. By definition of an angle bisector, angle BAD is congruent to angle DAC. Therefore, triangle PAD is similar to triangle PAB by the ASA postulate. As a result of this similarity, the ratio of the distances from P to line AB and from P to line AC must be equal, implying that these distances are the same.

Conversely, if a point P is such that its distances to lines AB and AC are the same, and it lies in the interior of angle BAC, triangles PAB and PAC are congruent by SAS postulate (since AP is a common side, PB=PC, and angle BAP is congruent to angle CAP due to equal distances implying perpendicular bisectors), hence angle PAB is congruent to angle PAC, and therefore, P must lie on the bisector of angle BAC. This completes the proof of the theorem, establishing the equivalence condition.