Question

Prove that if the 3 altitudes of a triangle are equal then the triangle is equilateral.

See attachment

Answer 1

I tried to do this and have got so far that AB = AC

Answer 2

@ParthKohli

Answer 3

@experimentX

Answer 4

I took following triangles :ABF and AEC
by AAS I got both of these triangles congruent

and therefore by CPCT AB = AC

Answer 5

(CPCT = Congruent Parts of Congruent Triangles are equal)

Answer 6

That idea should work to get that AB=BC as well.

Answer 7

So far , we have got ABC as isosceles.

Now, I took triangle ABC and AFC

both of these triangles are congruent by RHS

BF = FC ( by CPCT )

therefore F is mid point of BC
an AF is median and altitude also

Therefore , this is the property of an equilateral triangle , so ABC is an equi. triangle

Answer 8

Is this correct?

Answer 9

You mean to take ABE and BEC triangles @joemath314159 ?

Answer 10

yep. and do exactly what you did with the other triangles again.

Answer 11

I meant this : ADC and ABD in the above comment.

Answer 12

OK wait..

Answer 13

I think I had tried that earlier but I didnt' get any favorable result , wait let me do it again

Answer 14

there are many combinations you could take. The pair I see is triangles BEC and BDA.

Answer 15

Oh yes that will work ... BEC and BDA ..

this gives some output to me.

Answer 16

Thanks a lot @joemath314159 for your help ..

Might gonna work in more concentration for me..