isosceles triangle theorem


The following theorem holds in geometriesMathworldPlanetmath in which isosceles triangleMathworldPlanetmath can be defined and in which SSS, AAS, and SAS are all valid. Specifically, it holds in Euclidean geometry and hyperbolic geometry (and therefore in neutral geometry).

Theorem 1 ().

Let ABC be an isosceles triangle such that AB¯AC¯. Let DBC¯. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    AD¯ is a median

  2. 2.

    AD¯ is an altitudeMathworldPlanetmath

  3. 3.

    AD¯ is the angle bisectorMathworldPlanetmath of BAC

ABDC
Proof.

12: Since AD¯ is a median, BD¯CD¯. Since we have

  • AB¯AC¯

  • BD¯CD¯

  • AD¯AD¯ by the reflexive property (http://planetmath.org/ReflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath) of

we can use SSS to conclude that ABDACD. By CPCTC, ADBADC. Thus, ADB and ADC are supplementaryPlanetmathPlanetmath (http://planetmath.org/SupplementaryAngle) congruent angles. Hence, AD¯ and BC¯ are perpendicularMathworldPlanetmathPlanetmathPlanetmath. It follows that AD¯ is an altitude.

23: Since AD¯ is an altitude, AD¯ and BC¯ are perpendicular. Thus, ADB and ADC are right anglesMathworldPlanetmath and therefore congruent. Since we have

  • BC by the theorem on angles of an isosceles triangle

  • ADBADC

  • AD¯AD¯ by the reflexive property of

we can use AAS to conclude that ABDACD. By CPCTC, BADCAD. It follows that AC¯ is the angle bisector of BAC.

31: Since AD¯ is an angle bisector, BADCAD. Since we have

  • AB¯AC¯

  • BADCAD

  • AD¯AD¯ by the reflexive property of

we can use SAS to conclude that ABDACD. By CPCTC, BD¯CD¯. It follows that AD¯ is a median. ∎

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

Title isosceles triangle theorem
Canonical name IsoscelesTriangleTheorem
Date of creation 2013-03-22 17:12:12
Last modified on 2013-03-22 17:12:12
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 51-00
Classification msc 51M04
Related topic ConverseOfIsoscelesTriangleTheorem