Geometrical Proof: Prove Intersection Point on Line CM

[franceboy]

Homework Statement

Consider an triangle ABC with M as the middle point of the side AB.
On the straight line through AB you put the angle ∠ ACM at A and the angle ∠ MCB at B. Now you have two new lines. The new lines should be on the same side of AB as C.
Proof that the intersection point of the two new lines is located on the line through CM.

Homework Equations

The Attempt at a Solution

I wanted to use Ceva`s Theorem but I could not use it :(

I hope you can give me some advice.

 

[Joffan]

You've generated some similar triangles there which should help, if you work out the scaling. Note that |AM| = |BM|

 

[Mark44]

Homework Statement

Consider an triangle ABC with M as the middle point of the side AB.
On the straight line through AB you put the angle ∠ ACM at A and the angle ∠ MCB at B. Now you have two new lines.

I don't understand. From your explanation you would have two new angles. In my drawing below I have labelled ∠ ACM as a and ∠ MCB as b.

The new lines should be on the same side of AB as C.
Proof that the intersection point of the two new lines is located on the line through CM.

?

Homework Equations

The Attempt at a Solution

I wanted to use Ceva`s Theorem but I could not use it :(

What is Ceva's Theorem?

I hope you can give me some advice.

 

[Joffan]

Hi Mark, this is the construction as I understand it:

 

[Mark44]

Hi Mark, this is the construction as I understand it:
View attachment 77279

OK, that makes sense.

 

[franceboy]

Sorry thai i did not add a sketch but a GeoGebra sketch was not accepted as file.
Joffan your sketch is correct. I used the symmetry so that I " only" need to proof
BX / XC * CY / YA = 1
I determined all the angles and I found some similarities but they did not help to solve the problem.
Is Ceva' s theorem the right idea to solve the problem?

 

[Joffan]

Maybe you could use Ceva's theorem - it seems a little overpowered.

I would proceed by marking X as the intersection of the new line from A with CM and Y as the intersection of the new line from B with CM. Then show that |MX| = |MY| and thus that X == Y