Plane geometry with triangles and circles

Mark relationships before lengths

In a geometry description, a word such as tangent, diameter or parallel can be worth an entire extra equation. Start by naming the points and writing down what that word guarantees. Draw a rough sketch with room to label the segments. Its appearance need not be to scale. Reserve right-angle marks for a stated perpendicular or one that follows from a theorem. A triangle with a nearly horizontal side and a nearly vertical side in your sketch may have no right angle at all.

The three interior angles of a triangle sum to a straight angle. A pair of equal angles in two triangles, together with a second equal pair, fixes the third pair and proves similarity (similitudine). Similar triangles have one common scale factor for corresponding side lengths. Write the vertex names in matching order before forming a ratio. In particular, the side opposite the largest angle must match the side opposite the largest angle in the other triangle. This check catches a fraction written the wrong way round in just one of the two triangles.

Drop an altitude to the hypotenuse

In a right triangle, Pythagoras relates the squares of its legs to the square of its hypotenuse. The hypotenuse lies opposite the right angle, so locate it before inserting numbers. Now draw the perpendicular from the right angle to the hypotenuse. Each of the two smaller triangles is similar to the original because it has a right angle and shares an acute angle with the original triangle. They are therefore similar to each other as well.

Let the two pieces of the hypotenuse have lengths $p$ and $q$ and let the altitude have length $h$. Corresponding sides in the two smaller triangles give $p/h=h/q$, hence $h^2=pq$. Each original leg also satisfies a projection rule: its square equals the whole hypotenuse times the adjacent piece of the hypotenuse. Decide which piece touches the endpoint of the leg you need. The larger leg must correspond to the larger projection. These two square relations are often called Euclid's theorems (teoremi di Euclide) for a right triangle.

Name the intercepted arc

A chord (corda) joins two points on a circle. A central angle has its vertex at the centre. An inscribed angle has its vertex on the circumference and its sides meet the circle again at the chord's endpoints. Follow the arc between those endpoints which does not contain the inscribed angle's vertex. The inscribed angle measures half the central angle sweeping that same arc. A minor arc and a major arc can share exactly the same endpoints, so endpoints alone do not tell you which central angle to use.

A diameter intercepts a semicircle, whose central angle is a straight angle. The corresponding inscribed angle is therefore a right angle anywhere on the opposite semicircle, away from the endpoints. If the chord falls short of being a diameter, an inscribed triangle need not be right-angled. A tangent (tangente) touches the circle at one point and is perpendicular there to the radius. This gives another reliable way to establish a right angle without trusting the sketch.

Match the parts in a segment product

For two chords intersecting inside a circle, the product of the two pieces of one chord equals the product of the two pieces of the other. The relation follows by matching angles in two triangles formed by the intersecting chords. For a point outside the circle with a tangent and a secant, the tangent length squared equals the outside part of the secant times its whole length. Mark the far endpoint before adding the outside and inside pieces. Both sides of either product represent length squared. If the final question asks for a length after solving a square equation, take the positive square root and give ordinary length units.