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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
For two lines AD and BC that intersect each other at P and for which A, B, C, D all lie on the same circle, the following equation holds:
The theorem follows directly from the fact that the triangles △PAC and △PBD are similar. They share ∠DPC and ∠ADB = ∠ACB as they are inscribed angles over AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above:
Next to the intersecting chords theorem and the tangent-secant theorem, the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle – the power of point theorem.
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In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines AD and BC that intersect each other at P and for which A, B, C, D all lie on the same circle, the following equation holds: | P A | ⋅ | P D | = | P B | ⋅ | P C | {\displaystyle |PA|\cdot |PD|=|PB|\cdot |PC|} The theorem follows directly from the fact that the triangles △PAC and △PBD are similar. They share ∠DPC and ∠ADB = ∠ACB as they are inscribed angles over AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: P A P C = P B P D ⇔ | P A | ⋅ | P D | = | P B | ⋅ | P C | {\displaystyle {\frac {PA}{PC}}={\frac {PB}{PD}}\Leftrightarrow |PA|\cdot |PD|=|PB|\cdot |PC|} Next to the intersecting chords theorem and the tangent-secant theorem, the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle – the power of point theorem.
In Euclidean geometry, the tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's Elements. Given a secant g intersecting the circle at points G1 and G2 and a tangent t intersecting the circle at point T and given that g and t intersect at point P, the following equation holds: | P T | 2 = | P G 1 | ⋅ | P G 2 | {\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|} The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.
The constant chord theorem is a statement in elementary geometry about a property of certain chords in two intersecting circles. The circles k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} intersect in the points P {\displaystyle P} and Q {\displaystyle Q} . Z 1 {\displaystyle Z_{1}} is an arbitrary point on k 1 {\displaystyle k_{1}} being different from P {\displaystyle P} and Q {\displaystyle Q} . The lines Z 1 P {\displaystyle Z_{1}P} and Z 1 Q {\displaystyle Z_{1}Q} intersect the circle k 2 {\displaystyle k_{2}} in P 1 {\displaystyle P_{1}} and Q 1 {\displaystyle Q_{1}} . The constant chord theorem then states that the length of the chord P 1 Q 1 {\displaystyle P_{1}Q_{1}} in k 2 {\displaystyle k_{2}} does not depend on the location of Z 1 {\displaystyle Z_{1}} on k 1 {\displaystyle k_{1}} , in other words the length is constant. The theorem stays valid when Z 1 {\displaystyle Z_{1}} coincides with P {\displaystyle P} or Q {\displaystyle Q} , provided one replaces the then undefined line Z 1 P {\displaystyle Z_{1}P} or Z 1 Q {\displaystyle Z_{1}Q} by the tangent on k 1 {\displaystyle k_{1}} at Z 1 {\displaystyle Z_{1}} . A similar theorem exists in three dimensions for the intersection of two spheres. The spheres k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} intersect in the circle k s {\displaystyle k_{s}} . Z 1 {\displaystyle Z_{1}} is arbitrary point on the surface of the first sphere k 1 {\displaystyle k_{1}} , that is not on the intersection circle k s {\displaystyle k_{s}} . The extended cone created by k s {\displaystyle k_{s}} and Z 1 {\displaystyle Z_{1}} intersects the second sphere k 2 {\displaystyle k_{2}} in a circle. The length of the diameter of this circle is constant, that is it does not depend on the location of Z 1 {\displaystyle Z_{1}} on k 1 {\displaystyle k_{1}} . Nathan Altshiller Court described the constant chord theorem 1925 in the article sur deux cercles secants for the Belgian math journal Mathesis. Eight years later he published On Two Intersecting Spheres in the American Mathematical Monthly, which contained the 3-dimensional version. Later it was included in several textbooks, such as Ross Honsberger's Mathematical Morsels and Roger B. Nelsen's Proof Without Words II, where it was given as a problem, or the German geometry textbook Mit harmonischen Verhältnissen zu Kegelschnitten by Halbeisen, Hungerbühler and Läuchli, where it was given as a theorem.
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements. For example, for two chords AC and BD intersecting at point S, the following equation holds: | A S | ⋅ | S C | = | B S | ⋅ | S D | {\displaystyle |AS|\cdot |SC|=|BS|\cdot |SD|}
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Shirley Chisholm, elected in 1968.
