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Tagged as angle bisector, bisectors, incircle, integer, pythagoras, ratio, right triangle, sides
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I kept up with you until r = GC/(1+GC/b) and then couldn’t figure out how you simplified it to ab/(a+b+c). That is a very cool conjecture!
Thank you! And, you’re right: all Pythagorean triple triangles do have a whole number for the incircle radius.
Reblogged this on nebusresearch and commented:
For that “About An Inscribed Circle” problem I posted the other day: HowardAt58 worked out one way of proving what the radius of the circle that just fits within the 5-12-13 right triangle has to be, and in a pretty neat geometric fashion. Worth the read. I recommend following his steps along by hand, writing each out, but that reflects that I’m much more likely to follow mathematical reasoning if I write it out, even if I don’t do something past what the original author does. HowardAt58 also includes a little conjecture, inspired by playing around with a couple of Pythagorean triangles (playing around with a couple of examples is a great way to find conjectures), which I at least believe to be true.
Interestingly, his proof isn’t the same geometric proof that I’d realized we could do, so, I’m thinking to include that as another follow-up around here when I can make a couple diagrams that explain it.
Okay, now you’ve got me thinking about constructing these triangles. I’m up late and have to work tomorrow, but here is my first GeoGebra construction of a 3, 4, 5 with radius 1…the simplest one. I constructed the circle with radius 1, the lengths 3 and 4, and connected D and E and it was tangent to the circle. (I couldn’t figure out how to attach it. I’ll send it via email.)
I’m not that good at GeoGebra, but it would be interesting to show a series of circles with integer radii, r = 1, 2, 3, 4, 5… and see what the image would look like. Also show the measurements of the legs of each triangle.
How is the image of 6, 8, 10 related to 3, 4, 5 and 12, 16, 20?
How is the image of 5, 12, 13 related to the image of 3, 4, 5?
I would assume that there are families of images that have the same characteristics.
Just a thought…anyone up for the challenge? I’ll try it out when I have more time.
I am amazed at how your mind works. =) I learned all this in high school. Actually went to the top math/science hs in NYC. But my mind doesn’t work like this.
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