A worksheet on constructions, the RASS theorem, and inscribed/circumscribed circle to a triangle.

Warm-up: constructions and congruence.
(a)
Using only a compass and straightedge, construct the following:
  • the midpoint of a segment,
  • the bisector of an angle,
  • a perpendicular to a line through a given point,
  • a square on a given segment,
  • a line parallel to a given line through a given point.
(b)
Recall the triangle congruence theorems:
\[ \text {SAS, SSS, AAS/ASA, RASS.} \]
Puzzle.

Let \(A\), \(B\), and \(C\) be three non-collinear points.

Construct, using only a straightedge and compass, a circle passing through \(A\), \(B\), and \(C\).

Remark. This is possible in \(7\) moves.

Comparing chords via the center.

Let \(AB\) and \(CD\) be chords of a circle of center \(E\).

  • Consider the line \(\ell _1\) perpendicular to \(AB\) passing through \(E\), and call \(F\) the intersection point with \(AB\).
  • Consider the line \(\ell _2\) perpendicular to \(CD\) passing through \(E\), and call \(G\) the intersection point with \(CD\).

Show that

\[ EF = EG \quad \text {if and only if}\quad AB = CD. \]
Kites and perpendicular diagonals.

Let \(ABCD\) be a quadrilateral such that

\[ AB = AD \quad \text {and}\quad BC = CD. \]

Show that the diagonals \(AC\) and \(BD\) are perpendicular.

Angle bisectors in a triangle.

Show that all the angle bisectors in a triangle intersect at a common point.

Idea. Construct the angle bisectors at \(\angle A\) and \(\angle B\), and let \(D\) be their intersection point.

Goal. Show that \(DC\) bisects \(\angle C\).

Hint. Construct the perpendicular lines through \(D\) to:

  • \(BC\),
  • \(AB\),
  • \(AC\).

Using AAS, show that:

\[ FD = ED, \qquad ED = GD. \]

Conclude using

\[ FD = GD. \]
Bonus: classical constructions.

Using only a straightedge and compass, construct:

  • the circumscribed circle to a given triangle (par = \(7\)),
  • the inscribed circle to a given triangle (par = \(13\)).

How many moves did you use?