Geometric Construction 2

The locus of a point on the circumference of a circle rolling round the outside of a fixed circle is known as an epicycloid. In case the two circles have the same radius, the curve is called the cardioid. If the radius of the rolling circle is one-half that of the fixed circle, the curve is called the nephroid.
1. Construct the cardioid.
2. Construct the nephroid.
The locus of a point on the circumference of a circle rolling round the inside of a fixed circle is known as a hypocycloid. In case the rolling circles has a radius one-third that of the fixed circle, the curve is called the deltoid. If the radius of the rolling circle is one-fourth that of the fixed circle, the curve is called the astroid.
1. Construct the deltoid.
2. Construct the astroid.
Take any six points on the circumference of a circle and label them as
1. Construct the straight lines
2. Construct the intersection of the lines and Construct the intersection of the lines and Construct the intersection of the lines $31^{\prime }$ and $13^{\prime }.$
3. Show that no matter how the points are arranged, the points always lie on a straight line. This is known as Pascal's Theorem.
Given five points

we are to construct a conic passing through these points.
1. Construct the straight line
2. Construct the straight line .
3. Construct the intersection of the lines and
4. Construct the line
5. Construct the line $31^{\prime }.$
6. Construct a variable straight line (called the Pascal line) passing through the point
7. Construct the intersection of the lines and .
8. Construct the intersection of the lines $31^{\prime }$ and .
9. Construct the line connecting the points and $2^{\prime }.$
10. Construct the line $3^{\prime }1$ connecting and
11. Construct the intersection $3^{\prime }$ of the lines and $3^{\prime }1.$
12. The locus of $3^{\prime }$ , as varies, is the required conic.

Reference: H.S.M. Coxeter, The Real Projective Plane, p.91.

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