SOLUTION: given:triangle abc, ab=ac, d is the point between b and c. prove: ab>ad so far i wrote the 2 givens and the third statement which is <c=<d because if 2 ext sides are congruent

Question 826466: given:triangle abc, ab=ac, d is the point between b and c.
prove: ab>ad
so far i wrote the 2 givens and the third statement which is AD because of the substitution, and longer sides are across from longer angles. Now, i need to write from statement 4 to 7 explaining how i got to my last 8th prove but i don't know how and what to write. Help please.

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
There are two possibilities:

Segment AD is perpendicular to to segment BC. If this is true, then triangle ADB is a right triangle (since perpendiculars form right angles). And AD is a leg of this triangle and AB is the hypotenuse. Hypotenuse's are always the longest side of a right triangle (since it is opposite the largest angle).
It is unknown if AD is perpendicular to BC. In this case:
1. Draw a perpendicular from A to BC and label the point on BC as point E.
2. We now have two right triangles: AEB and AED.
3. Both right triangles share the leg AE.
4. EB > ED
5. AB is the hypotenuse of triangle AEB and AD is the hypotenuse of triangle AED.
6. Triangle AEB, with one leg identical and the other leg longer than triangle AED's, must have a longer hypotenuse.
7. So AB > AD.

Note: If point D is closer to point C than to point B then the logic above will not be as clear. But if you change the references from AB to AC then the logic still works. And, if AC > AD and AB = AC, then AB > AD.
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