First, I want to look at some examples to see if the sum of the ratios does equal 1 and 2 for some specific examples. Click HERE for a GSP file to explore.
Ok- I now have specific examples, but I need to prove it.
Part 1: I will begin with
Notice the denominators are the altitudes of the triangle ABC. If you multiply the altitude by the base and 1/2 you get the area of the triangle. So we will begin by multiplying each altitude by the corresponding base.
Now, each denominator is twice the area of triangle ABC. The numerators are the areas of the respective smaller triangles HBC, AHC, and AHB.
Since the denominators are the same I will add the numerators.
The three triangles in the numerator of the fraction will add up to the area of triangle ABC.
Which equals 1
Part 2: I will now take a look at the second part of the problem.
In part 1 I looked at areas to establish the ratios added up to 1. It would be nice if we could use the same type of reasoning to establish part 2. We have to think about how we could get the numerator in terms of altitudes, so we can find the areas of the triangles. Notice the numerators are part of the altitudes of triangle ABC; therefore we will re-write the numerators by subtracting part of the altitude from the entire altitude.
So AH = AD – HD, BH = BE – HE, and CH = CF – HF.
From there we will use the same reasoning as before and multiply each ratio by the length of the side that will give the area of the triangle.
The denominators are twice the area of triangle ABC. Notice, I am distributing in the numerator.
Notice the two lengths multiplied together in the numerators result in twice the area of triangle ABC.
The two lengths left in the numerators are twice the area of the smaller triangles. So,
Next, I simplify.
The triangles in the parenthesis add up to the area of triangle ABC.