ABC is an isosceles triangle in which AC = BC. AD and BE are respectively two altitudes to sides BC and AC. Prove that AE = BD.
Given: ABC is isosceles triangle AB = AC
AD and BE are altitudes at BC and AC
To Prove: AE = BD
Proof: In ABD & ACD
AD = AD (common)
ADB = ADC (AD is altitude at BC)
AB = AC (Given)
ABD ACD (by SAS congruence)
BAD = DAC (by CPCT)
BAD = DAE (ÐDAC = ÐDAE) (i)
Now, in ABD and ABE
AB = BA (common)
ADB = AEB = 90° (AD and BE are altitudes)
BAD = DAE (From i)
ABD ABE (by AAS congruency)
AE = BD (by CPCT)
Hence proved.