SOLUTION: prove that lengths of the perpendiculars from any point on the bisector of an angle to the sides are equal

Question 1100917: prove that lengths of the perpendiculars from any point on the bisector of an angle to the sides are equal

Answer by ikleyn(52800) (Show Source): You can put this solution on YOUR website!
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1.  Make a sketch.


2.  Find two right-angled triangles in your sketch.


3.  They are right angled triangles and have congruent acute angles, according to the condition.

    Hence, the other pair of their acute angles are congruent angles, too.


4.  These right-angled triangles have one side (the hypotenuse) common.


5.  So, you have two triangles with the common side,

    and the acute angles at the endpoints of this side are congruent, in pairs.


6.  Hence, the triangles are congruent, according to the ASA-test of triangle congruency.


7.  Then the two perpendiculars under the problem's question are congruent,

    as the corresponding sides of congruent triangles.


8.  It is what has to be proved, or QED in Latin.

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