Inner segment and square side

[edineireis]

how that the CP [edited] segment has the same measurement on the side of the square. M and N are midpoints.
Results already demonstrated: MPB angle is right.

I would like some tip to start the demonstration.
Thankful.
Edinei

 

[JeffM]

What in the world are you asking? CD is the side of the square.

There are isosceles, right, congruent, and similar triangles in that diagram. I might start by identifying which are which

 

[edineireis]

Sorry.
CP segment and not CD segment.

 

[Otis]

… I would like some tip to start the demonstration …

Hi edineireis. Are you using trigonometry or classic geometry?

Please post some of your work, if you can.

?

 

[edineireis]

Using classic geometry.

I managed to show two results:
- Triangle MPB is similar to triangle 3-4-5 - using circumferences centered on the vertices of the MPB triangle.
- It is possible to pass a circle through C, M, P and B, in which MB will be a diameter, using the right triangle theorem inscribed on the circle.

But I couldn't see a way to show that CP = CB.

 

[pka]

how that the CD segment has the same measurement on the side of the square. M and N are midpoints. Results already demonstrated: MPB angle is right. Edinei

Can you prove that \(\Delta ANB \cong \Delta CMB~?\)
Now you want to prove \(\Delta CPB\) is isosceles with \(\overline {CP} \cong \overline {CB} \)