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Prove angle $ADC$ is $120$ degrees
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Circles $A,B,C$ are all of equal radius $r$ and are all tangent to each other. A smaller circle $D$ is tangent to all three circles on the inside (as shown above)
I'm well aware that triangle $ABC$ is an equilateral triangle, and I realize that angle $ADC$ must be $120°$ but this is what I would like to prove.
I tried proving this by extending line segment $AD$ to the line $BC$ and making the intersection point $E$ and attempted to prove the two triangles $AEC$ and $AEB$ are congruent, which in turn would mean $AD$ is a bisector of the 60° angle $CAB$ but I ended up with SSA (side-side-angle), and this is the one that fails to prove triangles are congruent. I also don't think we necessarily know yet that extending this line would create perpendicular lines (do we?)
Another thought was to draw an angle bisector in, but how do we necessarily know that this new segment would go through point $D$?
How can I prove that this angle is $120°$ ?
geometry angle
asked Aug 1 at 17:49
WaveX
1,8111616
add a comment |Â
up vote
-2
down vote
favorite
Circles $A,B,C$ are all of equal radius $r$ and are all tangent to each other. A smaller circle $D$ is tangent to all three circles on the inside (as shown above)
I'm well aware that triangle $ABC$ is an equilateral triangle, and I realize that angle $ADC$ must be $120°$ but this is what I would like to prove.
I tried proving this by extending line segment $AD$ to the line $BC$ and making the intersection point $E$ and attempted to prove the two triangles $AEC$ and $AEB$ are congruent, which in turn would mean $AD$ is a bisector of the 60° angle $CAB$ but I ended up with SSA (side-side-angle), and this is the one that fails to prove triangles are congruent. I also don't think we necessarily know yet that extending this line would create perpendicular lines (do we?)
Another thought was to draw an angle bisector in, but how do we necessarily know that this new segment would go through point $D$?
How can I prove that this angle is $120°$ ?
geometry angle
asked Aug 1 at 17:49
WaveX
1,8111616
Do you see any symmetry in the figure?
– Batominovski
Aug 1 at 17:50
1
Use symmetry of circles and equilateral triangles. Just enough to say that the figure has triangular symmetry so the angle would be 120.
– Love Invariants
Aug 1 at 17:52
add a comment |Â
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
Circles $A,B,C$ are all of equal radius $r$ and are all tangent to each other. A smaller circle $D$ is tangent to all three circles on the inside (as shown above)
I'm well aware that triangle $ABC$ is an equilateral triangle, and I realize that angle $ADC$ must be $120°$ but this is what I would like to prove.
I tried proving this by extending line segment $AD$ to the line $BC$ and making the intersection point $E$ and attempted to prove the two triangles $AEC$ and $AEB$ are congruent, which in turn would mean $AD$ is a bisector of the 60° angle $CAB$ but I ended up with SSA (side-side-angle), and this is the one that fails to prove triangles are congruent. I also don't think we necessarily know yet that extending this line would create perpendicular lines (do we?)
Another thought was to draw an angle bisector in, but how do we necessarily know that this new segment would go through point $D$?
How can I prove that this angle is $120°$ ?
geometry angle
asked Aug 1 at 17:49
WaveX
1,8111616
Circles $A,B,C$ are all of equal radius $r$ and are all tangent to each other. A smaller circle $D$ is tangent to all three circles on the inside (as shown above)
I'm well aware that triangle $ABC$ is an equilateral triangle, and I realize that angle $ADC$ must be $120°$ but this is what I would like to prove.
I tried proving this by extending line segment $AD$ to the line $BC$ and making the intersection point $E$ and attempted to prove the two triangles $AEC$ and $AEB$ are congruent, which in turn would mean $AD$ is a bisector of the 60° angle $CAB$ but I ended up with SSA (side-side-angle), and this is the one that fails to prove triangles are congruent. I also don't think we necessarily know yet that extending this line would create perpendicular lines (do we?)
Another thought was to draw an angle bisector in, but how do we necessarily know that this new segment would go through point $D$?
How can I prove that this angle is $120°$ ?
geometry angle
asked Aug 1 at 17:49
WaveX
1,8111616
asked Aug 1 at 17:49
WaveX
1,8111616
asked Aug 1 at 17:49
WaveX
1,8111616
asked Aug 1 at 17:49
WaveX
1,8111616
1,8111616
Do you see any symmetry in the figure?
– Batominovski
Aug 1 at 17:50
1
Use symmetry of circles and equilateral triangles. Just enough to say that the figure has triangular symmetry so the angle would be 120.
– Love Invariants
Aug 1 at 17:52
add a comment |Â
Do you see any symmetry in the figure?
– Batominovski
Aug 1 at 17:50
1
Use symmetry of circles and equilateral triangles. Just enough to say that the figure has triangular symmetry so the angle would be 120.
– Love Invariants
Aug 1 at 17:52
Do you see any symmetry in the figure?
– Batominovski
Aug 1 at 17:50
Do you see any symmetry in the figure?
– Batominovski
Aug 1 at 17:50
1
1
Use symmetry of circles and equilateral triangles. Just enough to say that the figure has triangular symmetry so the angle would be 120.
– Love Invariants
Aug 1 at 17:52
Use symmetry of circles and equilateral triangles. Just enough to say that the figure has triangular symmetry so the angle would be 120.
– Love Invariants
Aug 1 at 17:52
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
Hint:
Connect points $B$ and $D$ with a line segment and observe that $triangle ADC$, $triangle BDC$, and $triangle ADB$ are congruent to each other.
answered Aug 1 at 17:53
Math Lover
12.2k21132
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hint:
Connect points $B$ and $D$ with a line segment and observe that $triangle ADC$, $triangle BDC$, and $triangle ADB$ are congruent to each other.
answered Aug 1 at 17:53
Math Lover
12.2k21132
add a comment |Â
up vote
0
down vote
Hint:
Connect points $B$ and $D$ with a line segment and observe that $triangle ADC$, $triangle BDC$, and $triangle ADB$ are congruent to each other.
answered Aug 1 at 17:53
Math Lover
12.2k21132
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Hint:
Connect points $B$ and $D$ with a line segment and observe that $triangle ADC$, $triangle BDC$, and $triangle ADB$ are congruent to each other.
answered Aug 1 at 17:53
Math Lover
12.2k21132
Hint:
Connect points $B$ and $D$ with a line segment and observe that $triangle ADC$, $triangle BDC$, and $triangle ADB$ are congruent to each other.
answered Aug 1 at 17:53
Math Lover
12.2k21132
edited Aug 1 at 18:05
answered Aug 1 at 17:53
Math Lover
12.2k21132
answered Aug 1 at 17:53
Math Lover
12.2k21132
answered Aug 1 at 17:53
Math Lover
12.2k21132
12.2k21132
add a comment |Â
add a comment |Â
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Do you see any symmetry in the figure?
– Batominovski
Aug 1 at 17:50
1
Use symmetry of circles and equilateral triangles. Just enough to say that the figure has triangular symmetry so the angle would be 120.
– Love Invariants
Aug 1 at 17:52