Math Is Fun Forum

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Geometry with Tangents

Let the incircle of triangle

be tangent to sides

,

, and

at

,

, and

, respectively. Prove that triangle

is acute.

Any help is appreciated, thanks!

Last edited by Enshrouded_ (2015-09-23 09:59:40)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Geometry with Tangents

hi Enshrouded_

Take a look at post 6 here http://www.mathisfunforum.com/viewtopic.php?id=17799

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Geometry with Tangents

Can you help a bit more? Thanks

Like, I understand the 5 properties now, but I'm stuck on how to use them.

EDIT Forgot a hint: Compute the angles of triangle $DEF$ in terms of the angles of triangle $ABC$.

Last edited by Enshrouded_ (2015-09-23 10:48:48)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Geometry with Tangents

hi Enshrouded_

Let O be the centre of the circle.

Let BAC = 2x, then angle AOE = AOF = 90 - x

So what is angle EOF in terms of x ?

What is FDE in terms of x ?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Geometry with Tangents

EOF would be 180-2x

And FDE would be 90-x

Okay, so I know FDE, but I just cant find out how to get FED and EFD. I can't find a way to get any more angle values that would help me.

Can you help fast? This problem is due extremely soon.

Last edited by Enshrouded_ (2015-09-24 13:00:34)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Geometry with Tangents

hi Enshrouded_

That's it.  You have all you need.

Whether 2x was acute or obtuse, you can be certain that x and also 90 - x are acute.

Now you can either do the same for angle ABC = 2y and ACB = 2z, or, it should be acceptable, say 'similarly' for ABC and ACB.  In a mathematical proof, where you have several similar cases, it is acceptable to prove once in detail, and then just say the other cases can be shown similarly.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob