We are given isosceles `Delta ABC` with vertex A, altitudes `bar(CE),bar(BD)` . We are asked to prove that `Delta HBC` is isosceles, where H is the intersection of the altitudes.
** Depending on the text and/or instructor, you may have to include more/less detail. **
(1) `bar(AB) cong bar(AC)` (1) Given
(2) `/_ABC cong /_ACB` (2) Isosceles triangle theorem *
(3) `/_CEB,/_BDC` are rt angles (3) Definition of altitude
(4) `/<CEB cong /_BDC` (4) Rt angles are congruent
(5) `bar(BC) cong bar(BC)` (5) Reflexive property of `cong`
(6) `Delta BEC cong Delta CDB` (6) AAS
(7) `/_DBC cong /_ECB` (7) CPCTC **
(8)`` `bar(HB) cong bar(HC)` (8) converse of isos. triangle thm*
(9) `Delta HBC` is isosceles (9) Definition of isosceles
* Isosceles triangle theorem -- If two sides of a triangle are congruent then the angles opposite those sides are congruent.
** CPCTC -- corresponding parts of congruent triangles are congruent
* Converse of isosceles triangle theorem -- If two angles of a triangle are congruent, then the sides opposite those angles are congruent