Find the measure of angle $MAN$.

Question

We construct two points $M,N$ such that perimeter of triangle $MCN$ is equal to half of the perimeter of the square.Find the measure of angle $MAN$.

The case $CM=CN$ leads to a simple equation so we can find the measure of $MAN$ using trigonometry.But in general case I don't no How to use equality of perimeter maybe we have to construct the same triangle in another place and use the sum of their perimeter.But I didn't get anything from this method.Any hints?

Answer 1

I think I understood about which square you say!

I think your problem is the following.

Let $ABCD$ is a square with $AB=a$. $M\in BC$ and $N\in CD$ such that $CN+CM+MN=2a$. Find $\measuredangle MAN$.

Let $MC=x$ and $CN=y$.

Hence, $$x+y+\sqrt{x^2+y^2}=2a,$$ $$AM=\sqrt{(a-x)^2+a^2}=\sqrt{2a^2-2ax+x^2}=\sqrt{\frac{1}{2}\left(x+y+\sqrt{x^2+y^2}\right)^2-\left(x+y+\sqrt{x^2+y^2}\right)x+x^2}=$$ $$=\sqrt{\sqrt{x^2+y^2}\left(y+\sqrt{x^2+y^2}\right)}.$$ By the same way we obtain: $$AN=\sqrt{\sqrt{x^2+y^2}\left(x+\sqrt{x^2+y^2}\right)}.$$ Thus, $$\measuredangle MAN=\arccos\frac{AM^2+AN^2-MN^2}{2AM\cdot AN}=$$ $$=\arccos\frac{4a^2-2a(x+y)+x^2+y^2-x^2-y^2}{2\sqrt{x^2+y^2}\sqrt{\left(y+\sqrt{x^2+y^2}\right)\left(x+\sqrt{x^2+y^2}\right)}}=$$ $$=\arccos\frac{2a(2a-x-y)}{2\sqrt{x^2+y^2}\sqrt{x^2+xy+y^2+(x+y)\sqrt{x^2+y^2}}}=$$

$$=\arccos\frac{a}{\sqrt{2a^2}}=\arccos\frac{1}{\sqrt2}=45^{\circ}.$$ Done!

Answer 2

By the beautiful Mick's hint we can get a nicer solution.

Let $P$ is a point inside our square $ABCD$ such that $PM=PN$, $\measuredangle MPN=90^{\circ}$

and segments $PC$ and $MN$ intersects.

Thus, $PMCN$ is cyclic and since $\measuredangle PCN=\measuredangle PMN=45^{\circ}$, we see that $P\in AC$.

Now, we'll use a rotation of $\Delta PCN$ around point $P$ on $90^{\circ}$.

Let $R^{90^{\circ}}_{P}\left(\Delta PCN\right)=\Delta{PKM}$.

Since $PMCN$ is cyclic, we obtain that $K$, $M$ and $C$ placed on common line $BC$.

In another hand, $\measuredangle KPC=90^{\circ}$.

Thus, $CP=KP=\frac{x+y}{\sqrt2}$ and $$AP=a\sqrt2-\frac{x+y}{\sqrt2}=\frac{2a-x-y}{\sqrt{2}}=\frac{\sqrt{x^2+y^2}}{\sqrt2}=MP=NP,$$ which says that $P$ is a circumcenter of $\Delta AMN$ and $$\measuredangle MAN=\frac{1}{2}\measuredangle MPN=\frac{1}{2}\cdot90^{\circ}=45^{\circ}$$ and we are done!