Wentworth's Plane and Solid Geometry · Book II, Props. XXXV–XXXVIII · Book 3: Book II: The Circle
35. Circles about and in triangles; tangents
Goal: Your student can circumscribe and inscribe a circle for a triangle, draw a tangent through a point, and construct a segment that holds a given angle.
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 protractor
What's in the book
Propositions XXXV–XXXVIII (§§ 314–318), the in-centre, circum-centre and escribed circles (§ 316), and Exercise 146.
The lesson, step by step
Warm up
Ask: What is the locus of points equidistant from the sides of an angle?
The new idea
To circumscribe a circle about a given triangle.
To inscribe a circle in a given triangle.
Through a given point, to draw a tangent to a given circle.
Prop. XXXVIII (§ 318): upon a given straight line, to describe a segment of a circle in which a given angle may be inscribed.
Draw the figure
Draw a large scalene triangle. Construct the perpendicular bisectors of two sides and draw the circle through the three corners. On a second triangle, bisect two angles, drop a perpendicular from the meeting point to a side, and draw the inscribed circle.
Walk through the proof
XXXV: the meeting point is equidistant from the three vertices (§ 160).
XXXVI: the meeting point of the bisectors is equidistant from the three sides (§ 162).
XXXVII: on OE as diameter draw a circle; it cuts the given circle at M, and angle OME is in a semicircle, so EM is perpendicular to the radius: a tangent.
XXXVIII: the tangent-chord angle equals the given angle, and every inscribed angle on that arc is measured by the same half-arc.
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and does each construction and explains why it works: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Ex. 146. To draw a tangent to a given circle, so that it shall be parallel to a given straight line.
Wrap up
Your student names the in-centre and circum-centre on the drawings (§§ 314–315 notes).
Answers to the book's problems
Ex. 146: draw the diameter perpendicular to the given line; the tangents at its ends are perpendicular to it, hence parallel to the line (§ 104). There are two.
Worked out for this site from the scan. If a number in your copy differs, trust the book.