Wentworth's Plane and Solid Geometry · Book VIII, Props. V–VIII · Book 9: Book VIII: The Sphere
120. Tangent planes; inscribed and circumscribed spheres
Goal: Your student can prove that a plane perpendicular to a radius at its end is tangent, that a sphere can be inscribed in and circumscribed about any tetrahedron, and that two spheres meet in a circle.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- A ball (an orange or a rubber ball) and a rubber band or string
What's in the book
Propositions V–VIII (§§ 765–777) with corollaries and definitions of inscribed and circumscribed spheres.
The lesson, step by step
Warm up
Ask: What line is tangent to a circle? (The perpendicular to a radius at its end.)
The new idea
A plane perpendicular to a radius at its extremity is tangent to the sphere.
A sphere may be inscribed in any given tetrahedron.
A sphere may be circumscribed about any given tetrahedron.
The intersection of two spherical surfaces is the circumference of a circle whose plane is perpendicular to the line joining the centres of the surfaces and whose centre is in that line.
Draw the figure
Set a ball on a table: the table is a tangent plane. Press two soap bubbles together (or imagine it): they meet in a circle.
Walk through the proof
V: any other point A of the plane is farther from O than P is (§ 512), so outside the sphere.
VI: the planes bisecting three dihedral angles meet at a point equally distant from all four faces (§ 559).
VII: the perpendiculars to two faces at the centres of their circumscribed circles meet at a point equally distant from all four vertices (§ 516).
VIII: turn the figure of two intersecting circles about the line of centres; the common point A traces a circle.
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Compare Props. VI and VII with the inscribed and circumscribed circles of a triangle in Book II. What is the same?
Wrap up
Ask: Where on a basketball resting on the floor does the floor touch it, and what is special about the radius to that point?