Πέμπτη, 20 Ιανουαρίου 2011

TRIANGLE CONSTRUCTION A, 2b+a, 2c+a

To construct triangle ABC if are given A, 2b + a = m, 2c + a = n

Analysis


Let ABC be the triangle in question.

We have:

m+n = 2(a+b+c) = 4s ==> the semiperimeter s is known

m-n = 2(b-c) ==> the difference b-c is known

Let E,D the points the a-excircle (Ia) touches AC,BC, resp.
The triangle DAIa has:
ADIa = 90 d., DAIa = A/2, AD = s. Therefore IaD = IaE = r_b is known.

Let M be the midpoint of BC. We have BIaC = 90-(A/2) and ME = (|b-c|)/2.(So the problem is eqivalent to construct triangle if are given:
A, b-c, r_b)

IaM^2 = IaE^2 + ME^2 = (r_b)^2 + ((b-c)/2)^2 ==> the median IaM is known.

In the triangle IaBC we know the angle Ia, the altitude and the median from Ia, therefore the problem is equivalent to construct triangle if are given:

A, h_a, m_a (altitude, median from A, resp.). This construction is left to the reader.

Exercises:

To construct triangle ABC if are given:

1. A, 2b - a = m, 2c + a = n

2. A, 2b - a = m, 2c - a = n

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