If two isosceles triangles have a common base, prove that the line joining their vertices bisects them at right angles.

Explanation

Given: `ABC` and `DBC` are two isosceles triangles with the common base `BC`.
To prove: The line joining their vertices bisects them at right angles.
Proof:
In `/_\ABD` and `/_\ACD`,
`AB=AC`
`BD=CD`
`AD=AD`
`/_\ABDcong/_\ACD` (`SSS` congruence rule)
`/_BAE=/_CAE` (CPCT)
In `/_\ABE` and `/_\ACE`
`AB=AC`
`/_BAE=/_CAE`
`AE=AE`
`/_\ABEcong/_\ACE` (`SAS` congruence rule)
`/_BEA=/_CEA`
`BE=EC` (CPCT)
`/_BEA+/_CEA=180^@` (Linear pair),br>`=>2/_BEA=180^@`
`=>/_BEA=90^@`
`=>/_BEA=90^@=/_CEA`
`therefore AD` bisects the base `BC` at right angles.
Hence proved.