# A quadrilateral inscribed in a rectangle

Given a rectangle $$ABCD$$ in which there is an inscribed quadrilateral $$XYZT$$, with exactly one vertex on each side of the rectangle, how could I prove that the perimeter of the inscribed quadrilateral is larger then $$2|AC|$$ (two diagonals)?

I tried to use the triangle inequality, but I can't find the right way to do it.

Let’s suppose that $$X$$, $$Y$$, $$Z$$, $$T$$ are in $$AB$$, $$BC$$, $$CD$$, $$DA$$, respectively. Construct the reflection $$X_1$$ of $$X$$ through $$AD$$, the reflection $$X_2$$ of $$X$$ through $$BC$$, and the reflection $$X_3$$ of $$X_1$$ through $$CD$$. Your diagram should look like this.

Now, we have $$XY+YZ+ZT+TX$$ $$=X_2Y+YZ+ZT+TX_1$$ $$\ge X_2Z+ZX_1$$ $$=X_2Z+ZX_3$$ $$\ge X_3X_2.$$ However, since $$X_1X_2=2AB$$, $$X_1X_3=2AD$$, $$\angle X_3X_1X_2=\angle DAB=90^\circ$$, $$\mathop{\bigtriangleup}\!X_1X_2X_3\mathrel{\sim}\mathop{\bigtriangleup}\!ABD,$$ so that $$X_3X_2=2AC,$$ and $$XY+YZ+ZT+TX\ge 2AC$$, as we wanted. $$\blacksquare$$

Just mirror your rectangle several times.

• The diagram could be better, but this is a very cool approach. Dec 27, 2019 at 6:10

Take first the special case where $$XYZT$$ is a parallelogram with sides parallel to diagonals $$AC$$, $$DB$$.

Then $$XY+ZT=2EF$$. And since $$\triangle ZCY$$ is right and $$F$$ bisects $$ZY$$, then $$ZY=2FC$$. Likewise $$TX=2AE$$.

Therefore perimeter $$P$$ of $$XYZT=2AC$$.

Now let $$X'$$ be any other point on $$AB$$. Join $$X'T$$ and $$X'Y$$, and through $$X$$ draw an ellipse with $$T$$, $$Y$$ as foci.

Since $$\triangle TAX\sim\triangle YBX$$, then$$\angle TXA=\angle YXB$$$$AB$$ is tangent to the ellipse at $$X$$ (see Apollonius, Conics III, 48), and all other points $$X'$$ on $$AB$$ lie outside the ellipse.

And since by the well-known property of an ellipse$$XT+XY=JT+JY=GH$$but$$JX'+X'Y>JY$$therefore$$X'T+X'Y>XT+XY$$Similarly, taking any other point Z' on $$CD$$, we show that$$Z'T+Z'Y>ZT+ZY$$Therefore, except in the special case first considered, in a quadrilateral inscribed in a rectangle$$P>2AC$$