Geometría básica

Art of Problem Solving

WOOT 2009–10 Geometry of the Triangle
1
1.1

Major Centers
The Centroid

A median of triangle ABC is a line joining the vertex of a triangle to the midpoint of the opposite side. The medians of a triangle concur at the centroid G.

A

G B 1.2 The Circumcenter C

The perpendicular bisectors of the sides of triangle ABC concur at the circumcenter O. Thecircumcenter is equidistant to the three vertices; in other words, OA = OB = OC. This common distance is called the circumradius R. The circumcircle of triangle ABC is the circle centered at O with radius R, and passes through all three vertices.

A

O B C

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Art of Problem Solving

WOOT 2009–10 Geometry of the Triangle
1.3 The Incenter

The (internal) angle bisectors of triangle ABC concur at the incenter I. The incenter is equidistant to the sides of the triangle. This common distance is called the inradius r. The incircle of triangle ABC is the circle centered at I with radius r, and is tangent to all three sides.

A

I B 1.4 TheOrthocenter C

The altitudes of triangle ABC concur at the orthocenter H.

A

H B 1.5 The Excenters C

In triangle ABC, the internal angle bisector of ∠A, the external angle bisector of ∠B, and the external angle bisector of ∠C concur at the A-excenter, denoted by IA . The A-excenter is equidistant to the sides of the triangle. This common distance is called the A-exradius, denoted by rA . TheA-excircle of triangle ABC is the circle centered at IA with radius rA , and is tangent to all three sides. The B, C-excenter, -exradius, and -excircle are defined similarly.

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Art of Problem Solving

WOOT 2009–10 Geometry of the Triangle

A IC IB C

IB

IA

1.6

Exercises

1. In triangle ABC, the medians AD, BE, and CF concur at the centroid G. (a) Triangle DEF is called the medial triangle of triangle ABC. Show that the medial triangle DEF is similar to triangle ABC. (b) Show that the areas of triangles AEG, CEG, CDG, BDG, BF G, and AF G are all equal. (c) Prove that AG : GD = 2 : 1. 2. In triangle ABC, the (internal) anglebisectors AX, BY , and CZ concur at the incenter I. (a) (Angle Bisector Theorem) Show that BX/XC = BA/AC. (b) Find ∠CIY in terms of ∠A, ∠B, and ∠C. 3. In triangle ABC, the altitudes AP , BQ, and CR concur at the orthocenter H. (a) (b) (c) (d) (e) Prove that quadrilateral QCP H is cyclic. Find ∠CHQ in terms of ∠A, ∠B, and ∠C. Show that triangles ABC, AQR, P BR, and P QC are similar. What is theorthocenter of triangle HBC? Triangle P QR is called the orthic triangle of triangle ABC. Prove that if triangle ABC is acute, then H is the incenter of orthic triangle P QR. What if triangle ABC is obtuse or right?

4. Show that I is the orthocenter of triangle IA IB IC .

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Art of Problem Solving

WOOT 2009–10 Geometry of the Triangle
2 Triangle Formulas

In this section, we establish important formulas involving the triangle. We will use the following conventional notation: • A, B, C – vertices and angles of triangle ABC • a = BC, b = AC, c = AB – side lengths of triangle ABC • ha , hb , hc – altitudes of triangle ABC • K = [ABC] – area oftriangle ABC • s = (a + b + c)/2 – semi-perimeter of triangle ABC • R – circumradius of triangle ABC • r – inradius of triangle ABC • rA , rB , rC – exradii of triangle ABC Following this notation in your solutions will make them much easier to read.

2.1

Area of a Triangle

We have the following formula for the area of a triangle: K= 1 aha 2 1 = ab sin C 2 = s(s − a)(s − b)(s − c)...