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Posts Tagged ‘Symmedian’

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In a , if two cevians and where are drawn such that , then and are said to be Isogonal to each other, i.e. they are called Isogonal lines.

The first theorem one should study regarding Isogonal Lines is the following:

Theorem 1: Steiner’s Theorem: In a , the two cevians are isogonal if and only if

Theorem 1: Steiner’s Theorem:

Proof of Steiner Theorem: I’ll prove the forward direction, the backward case is left to the interested reader. Let We have in , by the rule of sines, Similarly, by applying rule of sines in , Multiplying, we get, again, by the rule of Sines. This completes out proof. End of Proof of Steiner’s Theorem

Theorem 2: Isogonal Conjugate Theorem If in a the cevians are concurrent, and if are isogonal to respectively, then are concurrent. If concur at , and concur at , then is said to be the Isogonal Conjugate of . Therefore, one can restate the theorem as saying that every point has an Isogonal Conjugate, which is , of course, unique. Proof of Isgonal Conjugate Theorem: We have, by Steiner’s theorem,

Theorem 2: Isogonal Conjugate Theorem

Multiplying, we get, But, by Ceva’s Theorem, since are concurrent. Therefore, we have Therefore, are concurrent, by Converse of Ceva’s Theorem. End of Proof of Isogonal Conjugate Theorem

Now, it’s time to derive some important Corollaries from the above. But first we define what a symmedian is. A Symmedian is the cevian which the Isogonal Conjugate of the Median. Also, the three Symmedians of a triangle concur (From theorem 2) at the Symmedian Point (Also called the Lemoine Point).

Theorem 3: Corollary to Steiner’s Theorem In a , cevian is a symmedian if and only if Proof The proof follows directly from Steiner’s Theorem, by putting the ratio equal to . End of Proof

Theorem 4: Prove that in any triangle, the orthocentre and the circumcentre of those triangles are Isogonal Conjugates. (The proof, which involves very simple angle chasing, is left to the interested reader.)

Theorem 4:

Before going on to the next Theorem regarding Symmedians, we first define what we mean by the Intouch Triangle and the Gergonne Point. The Intouch Triangle is the triangle whose vertices are the points where the incircle of a triangle meets the sides of that triangle. The Gergonne Point is the point of concurrency of the lines joining a point of a triangle to the point where the incircle touches the side opposite to it. Thus, it’s the perspectrix of the main triangle and its Intouch Triangle.

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Archive by category 'Uncategorized'

Archive for the ‘Uncategorized’ Category

July 3rd, 2018 by Sarah Holt

Visiting a Bi-Rite Market, the Bi-Rite Creamery or attending an event with Bi-Rite Catering is a journey for each of your senses: looking, smelling, feeling, hearing, and of course, tasting. Our food is proof of the work we do to make sure every sense is engaged and satisfied. And all the certifications, awards, designations and kudos of the products we carry add up to one simple benefit: deliciousness.

Deliciousness is subjective. And that’s ok. In fact, that’s fantastic.

We’re deeply committed to sourcing the best, most sustainable products that are full of flavor so you can shop with confidence knowing that your next meal will be a memorable, delicious one. But if there’s ever a time when it’s not, for any reason, let us know. We promise we’ll make it right. Simply stop by, call, email or hit us up at @biritesf or Alexander McQueen Black Medium Lino Shopper Tote wz1kyW3z

Oh, and if you ever want a taste of something first, we’ll happily open it and taste it alongside you, and share more details about the thoughtful maker, rancher, or farmer who brought us those delicious eats.

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May 21st, 2018 by Renee Frojo

Sweet strawberries and tart rhubarb are a classic spring recipe combo, but the addition of creamy goat cheese and a toasted pecan crust adds a little bit of savory to mix things up.

This recipe calls for using both cooked and fresh strawberries, which are at their sweetest during peak season. Roasted strawberries make the tart nice and jammy, while the fresh strawberries and mint brighten the whole thing up.

Servings: 10 Prep Time: 1 hour 15 minutes Cook Time: 45 Minutes


½ cup pecans

1 cup plus 2 tablespoons all-purpose flour

3 teaspoons light brown sugar

½ teaspoon kosher salt

½ teaspoon ground cinnamon

½ cup (1 stick) chilled unsalted butter, diced

1 cup chopped rhubarb

¼ cup honey

1 pound strawberries, hulled and sliced into quarters

¼ cup sugar, plus more for sprinkling

¼ cup goat cheese, crumbled

2 tablespoons heavy cream

2 tablespoons mint leaves


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May 4th, 2018 by Renee Frojo

The best way to use up spring produce might just be to throw on it on a pizza.

For this recipe, buttery fava beans and bitter broccoli rabe are paired with creamy burrata cheese and a zesty pea shoot pesto. Half the toppings are cooked, while the other half are thrown on fresh, giving this pizza the best of both worlds.

Servings: 2 Prep Time: 15 minutes Cook Time: 15 Minutes


1 cup pea shoots ½ cup parsley ½ teaspoon salt 2 garlic cloves, sliced ½ cup pecorino cheese, grated ¼ cup pine nuts 1 teaspoon lemon juice ½ olive oil

1 cups shelled fava beans Kosher salt 2 teaspoons fresh lemon juice 3 tablespoons olive oil 1 pound Bi-Rite pizza dough Flour, for dusting 1 small bunch broccoli rabe 8 ounces burrata 4 ounces watercress 1 teaspoon Za’atar


First, make the pesto. In a food processor, blend the pea shoots, parsley, salt, garlic cloves, pecorino, pine nuts and lemon juice. As you blend, slowly add the olive oil into the mixture until smooth.

Next, cook the fava beans. In a medium saucepan, heat 3 cups water until boiling. Add fava beans and cook until tender, about 4 minutes. Drain and transfer to a bowl of cold water. Once cool enough to handle, peel the shells from the fava beans.

In a medium bowl, combine the fava beans with the lemon juice, 1 tablespoon oil, and a sprinkle of salt. Set aside.

Heat the oven to 450°F. Lightly flour a counter or cooking surface. Using your hands or a rolling pin, roll the dough out to a flattened disk. Place the dough on a greased baking sheet or baking sheet covered with parchment paper and then use your hands to pull the pizza dough into an oval shape.

Spoon a couple tablespoons of the pesto over the dough. Bake until the crust starts to brown, 8 to 10 minutes. Meanwhile, toss the broccoli rabe in the remaining oil and season with salt. Remove the pizza from the oven, spread half the burrata on top with the broccoli rabe. Place back in the oven until the broccoli rabe is cooked, 5 to 6 minutes. Remove from the oven, add the remaining burrata, fava beans, watercress and za’atar. Sprinkle with salt.

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The canonical work on designing programming systems for learning, and perhaps the greatest book ever written on learning in general, is Seymour Papert's "Mindstorms".

Mindstorms on Amazon Mindstorms at Powell's

Designing a learning system without a solid understanding of the principles in this book is like designing a mechanical system without understanding "the lever". Or "gravity". If you are reading this essay (and I'm pretty sure you are!) then you need to read "Mindstorms".

Seriously. I mean it. If you are going to design anything whatsoever related to learning, then you literally need to read "Mindstorms".

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This section will make reference to four seminal programming systems that were designed for learning, and I strongly recommend studying each of them.

Studying historical software can be challenging. It can be difficult to point to an accessible reference which explains a system's "heart". Here are the best references I can offer:

: we are very fortunate to have Seymour Papert's Mindstorms and The Children's Machine .

: I recommend Alan Kay's Ermanno Scervino Lace Flared Midi Dress Black AknC1GUp
and perhaps Dan Ingalls's Design Principles Behind Smalltalk .

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, but neither is as enlightening as a good analysis would be.

To be clear, I'm not advocating using any of these systems, in either their historical or modern incarnations. I'm advocating understanding them, and building on their insights.

In Logo , the programmer draws pictures by directing the "turtle", an onscreen character which leaves a trail as it moves:

Watch just two minutes of this video -- the children, and the beardy guy talking:

That's Seymour Papert explaining the Logo turtle. The turtle serves a number of brilliant functions, but the most important is that the programmer can identify with it. To figure out how to make the turtle perform an action, the programmer can ask how she would perform that action herself, if she were the turtle .

For example, to figure out how to draw a circle, a learner will walk around in circles for a bit, and quickly derive a "circle procedure" of taking a step forward, turning a bit, taking a step forward, turning a bit. After teaching it to herself, the learner can then teach it to the computer.* * Here, the learner has derived and implemented the differential equation for a circle, without knowing what a differential equation is. To quote Papert, a Logo program is an "intuitive analog of the differential equation". The turtle is the in-computer embodiment of the programmer herself, a "self", like the player-character in a video game, and thereby allows the learner to transfer her knowledge of her own body into knowledge of programming.

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