Incircle of a triangle
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In the above image, it says $$AE = frac{bc}{c+a}$$ and $$AF = frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ which implies $b=c$ which is absurd. What is wrong here?
geometry euclidean-geometry triangle recreational-mathematics plane-geometry
edited Jan 9 at 4:36
Andrei
11.7k21026
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
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add a comment |
$begingroup$
In the above image, it says $$AE = frac{bc}{c+a}$$ and $$AF = frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ which implies $b=c$ which is absurd. What is wrong here?
geometry euclidean-geometry triangle recreational-mathematics plane-geometry
edited Jan 9 at 4:36
Andrei
11.7k21026
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
$endgroup$
$begingroup$
If I'm not mistaken, $D$, $E$, $F$ are the points where the angle bisectors meet the sides of the triangle. They are not usually the points of tangency with the incircle. When the text discusses equilateral triangles, the points "become" the points of tangency.
$endgroup$
– Blue
Jan 9 at 4:25
$begingroup$
Just for curiosity, what book is it?
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– Dr. Mathva
Jan 9 at 21:05
$begingroup$
@Dr.Mathva Coordinate Geometry For JEE by Dr S K Goyal
$endgroup$
– Yash Chaudhary
Jan 10 at 3:21
add a comment |
$begingroup$
In the above image, it says $$AE = frac{bc}{c+a}$$ and $$AF = frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ which implies $b=c$ which is absurd. What is wrong here?
geometry euclidean-geometry triangle recreational-mathematics plane-geometry
edited Jan 9 at 4:36
Andrei
11.7k21026
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
$endgroup$
In the above image, it says $$AE = frac{bc}{c+a}$$ and $$AF = frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ which implies $b=c$ which is absurd. What is wrong here?
geometry euclidean-geometry triangle recreational-mathematics plane-geometry
geometry euclidean-geometry triangle recreational-mathematics plane-geometry
edited Jan 9 at 4:36
Andrei
11.7k21026
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
edited Jan 9 at 4:36
Andrei
11.7k21026
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
edited Jan 9 at 4:36
Andrei
11.7k21026
edited Jan 9 at 4:36
Andrei
11.7k21026
edited Jan 9 at 4:36
Andrei
11.7k21026
11.7k21026
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
asked Jan 9 at 4:12
Yash ChaudharyYash Chaudhary
112
112
$begingroup$
If I'm not mistaken, $D$, $E$, $F$ are the points where the angle bisectors meet the sides of the triangle. They are not usually the points of tangency with the incircle. When the text discusses equilateral triangles, the points "become" the points of tangency.
$endgroup$
– Blue
Jan 9 at 4:25
$begingroup$
Just for curiosity, what book is it?
$endgroup$
– Dr. Mathva
Jan 9 at 21:05
$begingroup$
@Dr.Mathva Coordinate Geometry For JEE by Dr S K Goyal
$endgroup$
– Yash Chaudhary
Jan 10 at 3:21
add a comment |
$begingroup$
If I'm not mistaken, $D$, $E$, $F$ are the points where the angle bisectors meet the sides of the triangle. They are not usually the points of tangency with the incircle. When the text discusses equilateral triangles, the points "become" the points of tangency.
$endgroup$
– Blue
Jan 9 at 4:25
$begingroup$
Just for curiosity, what book is it?
$endgroup$
– Dr. Mathva
Jan 9 at 21:05
$begingroup$
@Dr.Mathva Coordinate Geometry For JEE by Dr S K Goyal
$endgroup$
– Yash Chaudhary
Jan 10 at 3:21
$begingroup$
If I'm not mistaken, $D$, $E$, $F$ are the points where the angle bisectors meet the sides of the triangle. They are not usually the points of tangency with the incircle. When the text discusses equilateral triangles, the points "become" the points of tangency.
$endgroup$
– Blue
Jan 9 at 4:25
$begingroup$
If I'm not mistaken, $D$, $E$, $F$ are the points where the angle bisectors meet the sides of the triangle. They are not usually the points of tangency with the incircle. When the text discusses equilateral triangles, the points "become" the points of tangency.
$endgroup$
– Blue
Jan 9 at 4:25
$begingroup$
Just for curiosity, what book is it?
$endgroup$
– Dr. Mathva
Jan 9 at 21:05
$begingroup$
Just for curiosity, what book is it?
$endgroup$
– Dr. Mathva
Jan 9 at 21:05
$begingroup$
@Dr.Mathva Coordinate Geometry For JEE by Dr S K Goyal
$endgroup$
– Yash Chaudhary
Jan 10 at 3:21
$begingroup$
@Dr.Mathva Coordinate Geometry For JEE by Dr S K Goyal
$endgroup$
– Yash Chaudhary
Jan 10 at 3:21
add a comment |
1 Answer
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$begingroup$
$AE$ and $AF$ are not tangents. The angle bisector is not usually perpendicular to the opposite side. If that would be the case for all sides, then you have an equilateral triangle. It's easy to see in the figure that $AD$ is not perpendicular to $BC$
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
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add a comment |
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$begingroup$
$AE$ and $AF$ are not tangents. The angle bisector is not usually perpendicular to the opposite side. If that would be the case for all sides, then you have an equilateral triangle. It's easy to see in the figure that $AD$ is not perpendicular to $BC$
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
$endgroup$
add a comment |
$begingroup$
$AE$ and $AF$ are not tangents. The angle bisector is not usually perpendicular to the opposite side. If that would be the case for all sides, then you have an equilateral triangle. It's easy to see in the figure that $AD$ is not perpendicular to $BC$
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
$endgroup$
add a comment |
$begingroup$
$AE$ and $AF$ are not tangents. The angle bisector is not usually perpendicular to the opposite side. If that would be the case for all sides, then you have an equilateral triangle. It's easy to see in the figure that $AD$ is not perpendicular to $BC$
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
$endgroup$
$AE$ and $AF$ are not tangents. The angle bisector is not usually perpendicular to the opposite side. If that would be the case for all sides, then you have an equilateral triangle. It's easy to see in the figure that $AD$ is not perpendicular to $BC$
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
answered Jan 9 at 4:34
AndreiAndrei
11.7k21026
11.7k21026
add a comment |
add a comment |
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$begingroup$
If I'm not mistaken, $D$, $E$, $F$ are the points where the angle bisectors meet the sides of the triangle. They are not usually the points of tangency with the incircle. When the text discusses equilateral triangles, the points "become" the points of tangency.
$endgroup$
– Blue
Jan 9 at 4:25
$begingroup$
Just for curiosity, what book is it?
$endgroup$
– Dr. Mathva
Jan 9 at 21:05
$begingroup$
@Dr.Mathva Coordinate Geometry For JEE by Dr S K Goyal
$endgroup$
– Yash Chaudhary
Jan 10 at 3:21