# inscribe square into semicircle?

• Feb 15th 2009, 03:16 PM
Pabo258
inscribe square into semicircle?
How do you inscribe a square into a given semi-circle?
and i need the proof also.
• Feb 15th 2009, 03:43 PM
Nacho
Inscribe a rectangle of wide X and long 2X in a circunference...
• Feb 15th 2009, 05:32 PM
Soroban
Hello, Pabo258!

Quote:

How do you inscribe a square into a given semicircle?
Are you talking about a construction using Eucliden tools?

Here is one method . . .
Code:

```                        C * -                         /| :                         / | :                       /  | :           R  * * *  P/  | 2r           * - - - - o    | :         *  :        /:  * | :       *  :      / :  *| :           :      /  :    | :       *    :    /  :    * :       *----+----*----*----* -       B        O    Q    A       : -  r  - : -  r  - :```

We have a semicircle with center \$\displaystyle O\$ and radius \$\displaystyle OA = OB = r.\$

At \$\displaystyle A\$ erect a perpendicular to diameter \$\displaystyle AB\$
. . and measure off a distance \$\displaystyle 2r\!:\;AC = 2r\$

Draw line segment \$\displaystyle OC\$, intersecting the semicircle at \$\displaystyle P.\$

From \$\displaystyle P\$ drop a perpendicular \$\displaystyle PQ\$ to diameter \$\displaystyle AB.\$

From \$\displaystyle P\$ draw a horizotal line, intersecting the semicircle at \$\displaystyle R.\$

Segments \$\displaystyle PQ\$ and \$\displaystyle PR\$ determine the inscribed square.

I'll let someone else provide the proof . . .

• Jan 16th 2010, 05:44 AM
Garas
Proof
Does anyone know how to prove this?
• Jan 16th 2010, 04:32 PM
On Soroban's clever diagram

\$\displaystyle |AC|=2r\$

\$\displaystyle |OA|=r\$

\$\displaystyle |AC|=2|OA|\$

Triangles OPQ and OCA are similar.

Triangles OCA is a magnified version of triangle OPQ.

Therefore

\$\displaystyle |QP|=2|OQ|\$

Hence \$\displaystyle |QP|=|PR|\$ and the shape is a square.
• Jan 16th 2010, 04:46 PM