Which of these is a step in constructing an inscribed equilateral triangle using technology?

If the par score is 7, I'm afraid the best I've managed so far is this bogey 8!

Choose an arbitrary point $A$ on the given circle $\Gamma_0$, and an arbitrary radius, strictly less than the diameter of $\Gamma_0$. All the following circles $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$, $\Gamma_5$ are constructed with this radius.

1. Draw circle $\Gamma_1$, centre $A$, cutting $\Gamma_0$ at points $B$, $B'$.

2. Draw circle $\Gamma_2$, centre $B$, cutting $\Gamma_1$ at point $C$, on same side (of diameter through $A$) as $B'$.

3. Draw circle $\Gamma_3$, centre $B'$, cutting $\Gamma_1$ at point $C'$, on same side (of diameter through $A$) as $B$.

4. Draw circle $\Gamma_4$, centre $C$, cutting $\Gamma_3$ at point $D \ne A$.

5. Draw circle $\Gamma_5$, centre $C'$, cutting $\Gamma_2$ at point $D' \ne A$.

6. Draw $AD$, cutting $\Gamma_0$ at $E$.

7. Draw $AD'$, cutting $\Gamma_0$ at $E'$.

8. Draw $EE'$.

The triangle $AEE'$ is equilateral, and inscribed in $\Gamma_0$.

My justification of this construction (in rough, with a blunt pencil, on a very old sheet of graph paper, covered with previous failed attempts) is as follows:

Draw the equilateral triangle $ABC$, and its reflection on the other side of $AB$, whose apex is the other intersection (call it $F$) of $\Gamma_1$ and $\Gamma_2$; and similarly, the equilateral triangle $AB'C'$, and its reflection on the other side of $AB'$, whose apex is the other intersection (call it $F'$) of $\Gamma_1$ and $\Gamma_3$.

With the tangent to $\Gamma_0$ at $A$, the segments $AF$, $AC'$, $AB$, $AB'$, $AC$, $AF'$ make a series of angles: $$ \alpha + \left(\frac{\pi}{3} - 2\alpha\right) + 2\alpha + \left(\frac{\pi}{3} - 2\alpha\right) + 2\alpha + \left(\frac{\pi}{3} - 2\alpha\right) + \alpha = \pi. $$ By bisecting the two angles $2\alpha$, we construct two line segments making angles of $\pi/3$ with each other and with the tangent to $\Gamma_0$ at $A$. These suffice to construct the inscribed equilateral triangle.

I hope this sketch of a proof will be enough; it doesn't seem worth labouring, as it didn't make par.

1) Using the POINT TOOL & SEGMENT TOOL, label point C on circle A to create diameter CB. 2) Using the COMPASS TOOL, create a circle with radius AB and center point B 3) Using the POINT TOOL, mark points D and F where circle A intersects circle B. 4) Using the SEGMENT TOOL, draw a segment from point D to point F. 5) Using the SEGMENT TOOL, draw a segment from point D to point C. RESULT: Equilateral triangle DCF inscribed in circle A.

Page 2

Get the answer to your homework problem.

Try Numerade free for 7 days

Ohio State University

We don’t have your requested question, but here is a suggested video that might help.

What is the first step in constructing an inscribed square? A. Using a compass, construct a perpendicular bisector through the diameter of the circle. B. Draw any diameter in the circle. C. Draw lines to connect the four endpoints created by two diameters on the circle. D. Use a compass to mark arc lengths the length of the circle’s radius.