AD, BE, CF are equal altitudes of a a triangle ABC . . .

Question : AD, BE, CF are equal altitudes of a ΔABC. Prove that ΔABC is an equilateral triangle.

Doubt by Suraj 

Solution : 

Given : AD, BE and CF are altitudes of ΔABC.
AD=BE=CF
To Prove : ABC is an equilateral triangle. 

Proof :
Area of ΔABC
= ½×Base×Height
= ½×BC×AD — (1) 
Again, 
Area of ΔABC
= ½×Base×Height
= ½×AB×CF — (2) 

From equation (1) and (2) 
½×BC×AD=½×AB×CF
BC×AD=AB×CF
BC×AD=AB×AD [∵AD=CF]
BC=AB — (3)

Similarly 
BC=AC — (4)

From equation (3) and (4) 

AB=BC=CA

All sides of triangle ABC are equal. 

So, ABC is an equilateral triangle. 

Hence Proved. 

Note : The above question can also be proved by using the concept of congruency. Can you do that by yourself? 

Similar Question : In given figure the altitudes AD, BE and CF, the altitudes of triangle ABC are equal. Prove that ABC is an equilateral triangle.

Similar Question : The altitudes of ΔABC, AD, BE and CF are equal. Prove that ΔABC is an equilateral triangle.