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زكريا حسناوي
زكريا حسناوي
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About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, please do.

Let $ABCD$ be a quadrilateral whose vertices are not concyclic. For each vertex, consider the triangle formed by the other three vertices, and denote its circumcenter as follows:

$O_A$ is the circumcenter of $\triangle BCD$,

$O_B$ is the circumcenter of $\triangle ACD$,

$O_C$ is the circumcenter of $\triangle ABD$,

$O_D$ is the circumcenter of $\triangle ABC$.

Prove that the four points $O_A, O_B, O_C, O_D$ are not concyclic.

Prove that no circle passes through all four points $O_A, O_B, O_C, O_D$.

Also, if the property is known in advance, please add a source that mentions it.

enter image description here

About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, please do.

Let $ABCD$ be a quadrilateral whose vertices are not concyclic. For each vertex, consider the triangle formed by the other three vertices, and denote its circumcenter as follows:

$O_A$ is the circumcenter of $\triangle BCD$,

$O_B$ is the circumcenter of $\triangle ACD$,

$O_C$ is the circumcenter of $\triangle ABD$,

$O_D$ is the circumcenter of $\triangle ABC$.

Prove that the four points $O_A, O_B, O_C, O_D$ are not concyclic.

Prove that no circle passes through all four points $O_A, O_B, O_C, O_D$.

enter image description here

About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, please do.

Let $ABCD$ be a quadrilateral whose vertices are not concyclic. For each vertex, consider the triangle formed by the other three vertices, and denote its circumcenter as follows:

$O_A$ is the circumcenter of $\triangle BCD$,

$O_B$ is the circumcenter of $\triangle ACD$,

$O_C$ is the circumcenter of $\triangle ABD$,

$O_D$ is the circumcenter of $\triangle ABC$.

Prove that the four points $O_A, O_B, O_C, O_D$ are not concyclic.

Prove that no circle passes through all four points $O_A, O_B, O_C, O_D$.

Also, if the property is known in advance, please add a source that mentions it.

enter image description here

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زكريا حسناوي
زكريا حسناوي
  • 4.8k
  • 9
  • 26

About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, please do.

Let $ABCD$ be a quadrilateral whose vertices are not concyclic. For each vertex, consider the triangle formed by the other three vertices, and denote its circumcenter as follows:

$O_A$ is the circumcenter of $\triangle BCD$,

$O_B$ is the circumcenter of $\triangle ACD$,

$O_C$ is the circumcenter of $\triangle ABD$,

$O_D$ is the circumcenter of $\triangle ABC$.

Prove that the four points $O_A, O_B, O_C, O_D$ are not concyclic.

If you want a slightly sharper formulation (more typical for MSE), you could also phrase the conclusion as:

Prove that no circle passes through all four points $O_A, O_B, O_C, O_D$.

enter image description here

About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, please do.

$O_A$ is the circumcenter of $\triangle BCD$,

$O_B$ is the circumcenter of $\triangle ACD$,

$O_C$ is the circumcenter of $\triangle ABD$,

$O_D$ is the circumcenter of $\triangle ABC$.

Prove that the four points $O_A, O_B, O_C, O_D$ are not concyclic.

If you want a slightly sharper formulation (more typical for MSE), you could also phrase the conclusion as:

Prove that no circle passes through all four points $O_A, O_B, O_C, O_D$.

enter image description here

About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, please do.

Let $ABCD$ be a quadrilateral whose vertices are not concyclic. For each vertex, consider the triangle formed by the other three vertices, and denote its circumcenter as follows:

$O_A$ is the circumcenter of $\triangle BCD$,

$O_B$ is the circumcenter of $\triangle ACD$,

$O_C$ is the circumcenter of $\triangle ABD$,

$O_D$ is the circumcenter of $\triangle ABC$.

Prove that the four points $O_A, O_B, O_C, O_D$ are not concyclic.

Prove that no circle passes through all four points $O_A, O_B, O_C, O_D$.

enter image description here

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زكريا حسناوي
زكريا حسناوي
  • 4.8k
  • 9
  • 26

Non-concyclicity of the circumcenters of complementary triangles in a quadrilateral

About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, please do.

$O_A$ is the circumcenter of $\triangle BCD$,

$O_B$ is the circumcenter of $\triangle ACD$,

$O_C$ is the circumcenter of $\triangle ABD$,

$O_D$ is the circumcenter of $\triangle ABC$.

Prove that the four points $O_A, O_B, O_C, O_D$ are not concyclic.

If you want a slightly sharper formulation (more typical for MSE), you could also phrase the conclusion as:

Prove that no circle passes through all four points $O_A, O_B, O_C, O_D$.

enter image description here