In a dependence triangle between the parties and corners strictly connect also internal lines of a figure - height, a median and a bisector. Knowledge of these ratios significantly simplifies the solution of tasks.
Instruction
1. From three heights of a triangle of the smallest there will be that which is lowered on the biggest of the parties of a figure. To make sure of it, express all three heights of a triangle through the sizes of its parties and compare. Let from three parties of a, b, c of any acute triangle the party and — the greatest, the party with — the smallest. Let's designate ha the height lowered on the party and, hb height, which is carried out to the party of b, hc — height on side of the village. Height divides any triangle into two rectangular triangles in which this height will always be one of legs.
2. ha height which is carried out to the greatest party and, can be determined by Pythagorean theorem: ha² = b² - and ₁² or ha² = with² - and ₂². Where and ₁ and and ₂ — pieces into which the party and is divided by ha height. Also on Pythagorean theorem express two other heights of a triangle through its parties: hb² = a²-b ₁² or hb² = with²-b ₂²; hc²=a²-c ₁² or hc²=b²-c ₂².
3. From comparison of the formulas determining triangle heights it is obvious that the ratio between reduced and deductible gives the smallest difference in expressions of ha² = to b² - and ₁² and ha² = with²-and ₂² as deductible and ₁ and and ₂ — pieces of the greatest party of a triangle.
4. It is possible to determine the smaller height of a triangle also through a sine of the known corner of a triangle. If on a condition the greatest of corners is set, then this corner lies against the greatest party, and from it the smallest height is carried out. To avoid bulky calculations, express required height better through trigonometrical functions of two other corners of a triangle as the relation of the party of a triangle to a sine of an opposite corner — size for this triangle a constant. Therefore, the smallest height of a triangle of ha=b*SinB or ha=c*SinC, where In - a corner between the greatest party and and the party of b, and With — a corner between the greatest party and and the party from a triangle.