# isosceles triangle theorem

###### Theorem 1 ().
1. 1.

$\overline{AD}$ is a median

2. 2.

$\overline{AD}$

3. 3.

$\overline{AD}$$\angle BAC$

###### Proof.

$1\Rightarrow 2$: Since $\overline{AD}$ is a median, $\overline{BD}\cong\overline{CD}$. Since we have

• $\overline{AB}\cong\overline{AC}$

• $\overline{BD}\cong\overline{CD}$

• $\overline{AD}\cong\overline{AD}$ by the reflexive property () of $\cong$

we can use SSS to conclude that $\triangle ABD\cong\triangle ACD$. By CPCTC, $\angle ADB\cong\angle ADC$. Thus, $\angle ADB$ and $\angle ADC$ are supplementary  (http://planetmath.org/SupplementaryAngle) congruent angles. Hence, $\overline{AD}$ and $\overline{BC}$ are perpendicular    . It follows that $\overline{AD}$ is an altitude.

$2\Rightarrow 3$: Since $\overline{AD}$ is an altitude, $\overline{AD}$ and $\overline{BC}$ are perpendicular. Thus, $\angle ADB$ and $\angle ADC$ are right angles  and therefore congruent. Since we have

• $\angle B\cong\angle C$ by the theorem on angles of an isosceles triangle

• $\angle ADB\cong\angle ADC$

• $\overline{AD}\cong\overline{AD}$ by the reflexive property of $\cong$

we can use AAS to conclude that $\triangle ABD\cong\triangle ACD$. By CPCTC, $\angle BAD\cong\angle CAD$. It follows that $\overline{AC}$ is the angle bisector of $\angle BAC$.

$3\Rightarrow 1$: Since $\overline{AD}$ is an angle bisector, $\angle BAD\cong\angle CAD$. Since we have

• $\overline{AB}\cong\overline{AC}$

• $\angle BAD\cong\angle CAD$

• $\overline{AD}\cong\overline{AD}$ by the reflexive property of $\cong$

we can use SAS to conclude that $\triangle ABD\cong\triangle ACD$. By CPCTC, $\overline{BD}\cong\overline{CD}$. It follows that $\overline{AD}$ is a median. ∎

Remark: Another equivalent (http://planetmath.org/Equivalent3) condition for $\overline{AD}$ is that it is the perpendicular bisector  of $\overline{BC}$; however, this fact is usually not included in the statement of the Isosceles Triangle Theorem.

Title isosceles triangle theorem IsoscelesTriangleTheorem 2013-03-22 17:12:12 2013-03-22 17:12:12 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Theorem msc 51-00 msc 51M04 ConverseOfIsoscelesTriangleTheorem