how do you write this indirect proof?

Question

how do you write the following by using an indirect proof?

if a triangle is not isosceles, then the measure of the median to any side of the triangle is greater than the measure of the altitude to that side.

Answer 1

1. Triangle ABC is not isosceles. 1 Given.
2 Hence AB not = BC not = CA 2. Def. of scalene triangle.
3. From vertex C draw a altitude perpendicular to AB at D and a median to the midpoint of AB at E. 3. Construction.
4. Assume Median CE < altittude CD 4 trichotommy law.
5. CE is the hypotenuse of a right triangle 5. Construction
6. Thus CE > CD. 6. Hypotenuse >either leg.
7. This contradicts our assumption and hence CE >CD.

The same proof applies no matter whic vertex is chosen.

Answer 2

When the triangle is not isosceles, the foot of the altitude and the end-point of the median on the side will be distinct points. The part of the side between the the altitude and the median will form a right triangle in which median being the hypotenuse is the longest side and is longer than the altitude.

Answer 3

This is not a formal proof - more observational.

Isosceles triangles (including equilateral triangles) have at
least one median that is also an altitude, because the line
from the particular vertex intersects the opposite side at
right angles.

In general, from any given point, the shortest distance to
any given line (that does not contain the point), is in that
direction which makes both lines perpendicular to each other.

If a triangle is not isosceles, then it must be scalene, and
therefore has three unequal sides. In such triangles, the
medians are not concurrent with the altitudes.

The shortest distance from each vertex to the opposite side
is the altitude, which is perpendicular to the opposite side.
All other lines, which include the medians, are longer,
because none of them intersect perpendicularly.

Answer 4

A direct proof is of the form "P then Q". An indirect proof (aka proof by contradiction) is a logically equivalent statement which assumes "not Q" then proceeds to show "not P" by deriving a contradiction somewhere in the process. This is a very powerful method in many branches of math specially in geometry and number theory. So let's assume that the circle and the line do intersect (this is your "not Q" in the context of the given problem -- note that the negative of a negative statement is positive in a two-valued logic system). Hence there exists a point (x, y) such that the statements x^2 + (y - 4)^2 = 25 and y = 10 hold true at the same time. Now substitute y = 10 into the equation of the circle. At this point we obtain x^2 + (10 - 4)^2 = 25, or x^2 + 36 = 25, or x^2 = - 11. But this is impossible since x^2, a real perfect square, is always non-negative. So then the point (x, y) does not satisfy both of the two given defining equations at the same time. This establishes "not P" and our indirect proof is complete. //