3 lines coincide — Isoceles: median, altitude, bisector coincide · Principia

Dependencies: SAS congruence, Linear pair sums to 180°, and Protractor axiom (Angle bisector exists and is unique).

Statement

Let $\triangle ABC$ satisfy $AB = AC$ (i.e. isosceles, with $A$ the apex and $BC$ the base). Then, starting from $A$ , the bisector of the apex angle, the median to the base, and the altitude to the base are one and the same segment; if we call its endpoint on the base $D$ , then $D$ simultaneously satisfies

\angle BAD = \angle CAD,\qquad BD = DC,\qquad AD \perp BC.

$Three lines coincide: in isosceles \triangle ABC, the single segment AD from A serves at once as angle bisector, median, and altitude$

Isoceles: median, altitude, bisector coincide

Statement

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