Regular means all internal angles and all sides are equal.
Bob
]]>Can you answer me one question please? I know that a regular polygon is convex. But if the converse of this true? That is, a convex polygon is regular (in general)?
]]>I found two cases, x=36 and x=45. I think that's all.
At school I was taught geometry from a variety of books. I don't have a preferred one now. You could always refer to the master:
http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
Bob
]]>Also, please tell me a name for a book/eBook that can give me a detailed oriented topics regarding plane figures only e.g. Polygons, their types, properties, theorems regarding them etc.
]]>This is a general diagram. Lengths are not meant to be accurate. I have, however, bisected ABC accurately.
I have marked the two bisected halves with an x.
When I first read the problem, I was not sure which sides were equal in the isosceles triangles. I'm assuming that ABD and DBC are the isosceles triangles, but, for each, which are the two equal sides.
After a moment I realised that some possibilities are obviously impossible, so I decided to try all possibilities and find out which are impossible and which lead to answers. Here is my analysis:
case 1
AB = DB and DB = CB.
=> BAD = BDA = BDC = BCD = 90 - x/2
So at D, 90 - x/2 + 90 - x/2 = 180 => x = 0
So this case is not possible.
case 2
AB = DB and BD = DC
So DCB = x and so BDA = 2x.
Also BAD = 90 - x/2, so in triangle ABD
x + 90 - x/2 + 2x = 180 => 5x/2 = 90 => 5x = 180 => x = 36.
case 3
AB = BD and BC = DC
=> BDC = x and BDA = 90 - x/2
=> x + 90 - x/2 = 180 => x/2 = 90 => x = 180
So this case is not possible.
case 4
AB = AD and BC = DC
=> ADB = BDC = x => x = 90.
This case is not possible.
case 5
AB = AD and BD = DC
=> ADB = ACB = x => 2x = x => x = 0
This case is not possible.
I think that concludes all possible cases. Unless you can spot any others
Whoops. Found one myself.
case 6
AD = BD and BD = DC
So BAD = BCD = x and ADB = BDC = 2x => 4x = 180 => x = 45
Bob
]]>I haven't got that book. I'm having trouble getting a diagram. Please could you post one, or failing that, describe it using A, B, C etc.
Thanks,
LATER EDIT. Actually, I think I've narrowed it down to one case, but I'd still like a diagram to help confirm it.
Bob
]]>Question 157 of Kiselev's Book 1, Compute angles of a triangle which is divided by one of its bisectors into two isosceles triangles. Find all solutions.
]]>Hhmmm. It does lose something in the translation. I'm puzzled too. Does it mean this:
AB is congruent to DC. BF is congruent to CG.
The second lines are added to the first. Is AF congruent to DG?
Bob
]]>I can't understand the statement. For example, "the sums of two congruent line segments are non congruent (to what?)
Please help me understand the statement. I'm confident I can solve it myself. If you've ideas, please share.
]]>In combination, I've two more book as well, that I keep parallel while studying geometry. But if you know some better book(s), please share with me, I feel much pleasure when I get some interesting books.
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