Prove that the plane $(A'MN)$ is tangent to the sphere inscribed in the cube.

Question

question

Let the cube be $ABCDA'B'C'D'$, and the points $M$ and $N$ are the midpoints of the edges $(AB)$ and $(AD)$ respectively. Prove that the plane $(A'MN)$ is tangent to the sphere inscribed in the cube.

drawing

my idea

All I know about the sphere inscribed in the cub is that its tangent to the every face of the cube in the center of it.

So simply we have to show the plane given intersects the sphere only in one point.

I let M' and N' be the centres of the faces $A'B'BA$ and $ADD'A'$, both of them $\in$ the sphere as I said in the start.

We can show by the midline theorem that $MN \parallel M'N'$.

I don't know how much this is going to help, but this is all I could get from this problem, that makes sense. I don't know how to demonstrate it because I never saw hoe to demonstrate something is tangent with something.

(also if you think of a coordinate solution, i don't know this side of math)

Hope one of you can help me! Thank you!

Answer 1

Suppose, WLOG, the side of your cube is $2$ units. Then the equation of the plane is $$ 2x+2y+z=2. $$ The distance from that plane to the point $(1,1,1)$ is $$ |2\cdot 1+2\cdot 1+1-2|/\sqrt{2^2+2^2+1^2}=1, $$ which is precisely the radius of the sphere. Ergo, the plane is tangent to the sphere.

Answer 2

Consider the section $ACC'A'$ of your cube (figure below), where line $A'F$ is the intersection of planes $A'MN$ and $ACC'A'$, so that $AF={1\over4}AC={\sqrt2\over4}AA'$.

If $H$ is the midpoint of $AA'$ and $I$ is the midpoint of $AH$, then triangles $OHI$ and $A'AF$ are similar, because $$ {HI\over OH}={\sqrt2\over4}={AF\over AA'}. $$ It follows that $\angle OIH=\angle A'FA$ and triangles $OHI$, $A'GI$ are also similar (where $G=OI\cap A'F$), so that $\angle A'GI=90°$ and $OI\perp A'F$.

To complete the proof we must show that $$ OG=OI-IG={1\over2}AA'=\text{radius of the circle}. $$ In fact: $$ OI=\sqrt{OH^2+HI^2}={3\over4}AA', \quad IG={A'I\cdot AF\over A'F}={1\over4}AA'. $$